Adiabatic evolution and shape resonances
AAdiabatic evolution and shape resonances
Michael Hitrik
Department of MathematicsUniversity of CaliforniaLos Angeles, CA 90095-1555, USA [email protected]
Andrea Mantile
Laboratoire de Math´ematiques - FR3399 CNRSUniversit´e de ReimsMoulin de la Housse - BP 103951687 Reims, France [email protected]
Johannes Sj¨ostrand
IMB - UMR5584 CNRS , Universit´e de Bourgogne9, avenue Alain Savary - BP 4787021078 Dijon cedex, France [email protected]
Abstract
Motivated by a problem of one mode approximation for a non-linear evolution with charge accumulation in potential wells, we con-sider a general linear adiabatic evolution problem for a semi-classicalSchr¨odinger operator with a time dependent potential with a well inan island. In particular, we show that we can choose the adiabaticparameter ε with ln ε (cid:16) − /h , where h denotes the semi-classicalparameter, and get adiabatic approximations of exact solutions overa time interval of length ε − N with an error O ( ε N ). Here N > While deciding the general strategy through joint discussions, the coauthors haveinvested various amounts of time in the actual elaboration. The main authors of the a r X i v : . [ m a t h - ph ] N ov ´esum´e Motiv´es par un probl`eme d’approximation `a un mode pour une´evolution avec accumulation de charge dans des puits de potentiel,nous consid´erons un probl`eme d’´evolution lin´eaire pour un op´erateurde Schr¨odinger avec un potentiel d´ependant du temps avec un puitsdans une ˆıle. En particular, nous montrons que nous pouvons choisirle param`etre adiabatique ε avec ln ε (cid:16) − /h , o`u h d´esigne param`etresemi-classique, et obtenir des approximations adiabatiques de solu-tions exactes sur des intervalles de temps de longueur ε − N avec uneerreur O ( ε N ). Ici N >
Contents ε . . . . . . . . . . . . . . . . 33 H (Λ υG ) spaces 57 different sections are in indicated by their initials as follows:Section 1: MH, AM, JS,Sections 2, 3: AM, JS,Sections 4, 5, 6, 7: JS,Section 8: MH, JS,Section 9: MH, AM, JS. Our work is connected with the modelling of the axial transport through res-onant tunneling structures like highly doped p-n semiconductor heterojunc-tions (Esaki diodes), multiple barriers or quantum wells diodes. The scat-tering of charge carriers in such devices has been described using non-linearSchr¨odinger-Poisson Hamiltonians with quantum wells in a 1D semiclassicalisland (see [19]). The quantum wells regime is defined as a perturbation ofthe semiclassical Laplacian − h ∂ x by the superposition of a potential bar-rier plus an attractive term, with support of size h , modelling one or more quantum wells . In the simplest setting of a single well separating two lin-ear barriers, the linear part of the potential has the shape in Figure 1. Inconnection with the modelling of a mesoscopic semi-conductor device, thisscheme represents “metallic conductors” at ] − ∞ , a ] and [ d, + ∞ [ while thedouble barrier describes the interaction of charge carriers in a semiconduc-tor junction. Here a < b < d are fixed, c = b + h with h →
0, while V h corresponds to a rescaled Fermi length fixing the quantumscale of the system (see for instance [5]) and, coherently with the featuresof the physical model, is assumed to be small. The shape resonances (i.e.those with energies below V ) define the Fermi levels of the junction and thecorresponding resonant states describe (in the one-particle approximation)the concentration of charges in the depletion region. In particular, the exte-rior potential bias − V is introduced in order to select only incoming waveswith positive momentum as contributions to the charging process (about thispoint the reader may refer to the analysis developed in [6]).In linear models, the small- h asymptotic behaviour of the shape reso-nances generated by quantum wells has been understood in the work of B.Helffer and J. Sj¨ostrand [16]; for operators of the form H h := − h ∂ x + V h (1.1)with V h fulfilling the scaling of Figure 1 (i.e.: a semiclassical barrier sup-ported on [ a, d ] plus quantum wells) and suitable regularity assumptions, theapproach of [16] allows to localize the shape resonances w.r.t. the spectrumof a corresponding Dirichlet operator H hD := − h (cid:52) D ( a,d ) + 1 ( a,b ) V h , (1.2)where (cid:52) D ( a,d ) denotes the Dirichlet Laplacian on the barrier interval. In par-ticular, very accurate Agmon-type estimates show that to each λ ∈ σ p (cid:0) H hD (cid:1) ∩ (0 , V ) corresponds a unique resonance of H h , E h res = E h − i Γ h , and the esti-mates (cid:12)(cid:12) λ − E h (cid:12)(cid:12) + Γ h (cid:46) e − S /h , (1.3)hold with Γ h > E h res and S > quasiresonant states , which are L -functions defined by a cut-off of the resonant states outside the interactionregion (see the definition in [13]). This general idea has been investigatedin [36] in the framework of 3 D Schr¨odinger operators with exponentially de-caying potentials (see also [37] and [25]); for operators exhibiting the scalingintroduced above, a precise exponential decay estimate has been provided in[13, Th. 4.3]. Let u E h res denote the resonant state of H h for the resonance E h res (i.e.: a solution of (cid:0) H h − E h res (cid:1) u E h res = 0); under the assumption Γ h (cid:38) e − S /h (which holds for a large class of models including the case of sharp barriers(see [7])) we have e − itH h ( a,d ) u E h res = e − itE h res ( a,d ) u E h res + R h ( t ) , (1.4)4here R h ( t ) = O (cid:0) e − S /h (cid:1) , in the L -norm sense, on the time scale: t (cid:46) / Γ h .Then the estimate (1.3) implies (cid:12)(cid:12)(cid:12) e − itH h ( a,d ) u E h res (cid:12)(cid:12)(cid:12) ≈ e − t Γ h (cid:12)(cid:12) ( a,d ) u E h res (cid:12)(cid:12) . (1.5)The comparison between the shape resonance problem for H h and the eigen-value equation for the corresponding truncated Dirichlet model H hD alsoshows that the quasiresonant states are mainly supported near the wells (see[16]). Hence, according to the above relation, the time evolution preservesthis concentration of L -mass on the time scale 1 / Γ h which is exponentiallylarge w.r.t. h .In the non-linear modelling, the repulsive effect due to the concentrationof charges in the depletion region is taken into account by a Poisson potentialterm depending on the charge density. The corresponding non-linear steadystate problem (cid:0) H h + V hNL − E (cid:1) u = 0 , ∂ x V hNL = | u | , (1.6)has been investigated in [6]-[7] under far-from-equilibrium assumptions; inthis case, following the scaling introduced above, the underlying linear model H h is defined by using an array of quantum wells of the form W h = − N (cid:88) n =1 w n (( x − x n ) 1 /h ) , w n ∈ C ( R , R + ) , supp w n = [ − d, d ] . (1.7)An accurate microlocal analysis of the tunnel effect as h → { x n } . In the time-dependent case, the non-linear evolutionequation reads as i∂ t u = (cid:0) H h + V hNL (cid:1) u , ∂ x V hNL = | u | . (1.8)When the initial state is formed by a superposition of incoming waves withenergies close to the Fermi level ( E F the resonant energy), this interacts withresonant states which, as the estimates (1.5) suggest, are expected to evolvein time according to a quasi-stationary dynamics. In this picture, u behaveas a metastable state and the charge density | u | remains concentrated in aneighbourhood of the wells for a large range of time fixed by the imaginary5igure 2: The potential with charge accumulationpart of the (nonlinear) resonances. Depending on the position of the wells,this possibly induces a local charging process; then, the nonlinear couplingin (1.8) generates a positive response (depending on the charge in the wells)which modifies the potential profile and reduces the tunnelling rate.The above scheme outlines the behaviour of the nonlinear dynamics undernon-equilibrium initial conditions and assuming h small. In particular, V hNL isexpected to define an adiabatic process with variations in time of size ε = Γ h .The relevance of adiabatic approximations in the small- h asymptotic analysisof the nonlinear quantum transport was pointed out in ([19], [29], [30]) wherethis dynamics was considered within a simplified framework. Following the work [19] (with G. Jona-Lasinio), in [29], [30] C. Presilla and J.Sj¨ostrand considered a non-linear evolution problem for a mesoscopic semi-conductor device and did some heuristic work. It is assumed that the incom-ing charged particles (entering from the left) have energies E ≥ g ( E ) dE supported on [0 , E F ], where E F < V is theFermi level. Moreover, these particles interact only inside the device (i.e. inthe region ] a, d [ in Figure 1) through a modification of the common potentialdue to charge accumulation there. After a rescaling, the model is described6y a nonlinear Hamiltonian H hNL = H h + s (cid:0) u h ( t, · ) (cid:1) W ( x ) , where the linear part H h is defined as in (1.1) and Figure 1, while the Poissonpotential is replaced by an affine function W ( x ), with a fixed “profile” andsupport in ] a, d [, multiplied by the charge accumulated inside the device. Thisis defined in terms of the nonlinear evolution of generalized eigenfunctions u ( t, x, E ) according to s ( u h ( t, · )) := (cid:90) (cid:90) ( c + d ) / a + b ) / (cid:12)(cid:12) u h ( t, x, E ) (cid:12)(cid:12) g ( E ) dxdE , (1.9)and the corresponding nonlinear evolution problem is i∂ t u h ( t, x, E ) = (cid:0) H h + s ( u h ( t, · )) W ( x ) (cid:1) u h ( t, x, E ) , (cid:0) H h + s ( u h (0 , · )) W ( x ) − E (cid:1) u h (0 , x, E ) = 0 . (1.10)The heuristic analysis in [29] was based on the 1-mode approximation u h ( t, x, E ) ≈ µ ( t, x, E ) + e − iEt/h z h ( t, E ) e h ( x, s (cid:0) u h ( t, · ) (cid:1) ) , (1.11)where • µ ( t, x, E ) is the solution of a linear evolution problem, obtained by“filling” the potential well [ b, c ], • e h = e h (cid:0) x, s (cid:0) u h ( t, · ) (cid:1)(cid:1) is a resonant state ( (cid:54)∈ L ) corresponding to aresonance λ h ( s (cid:0) u h ( t, · ) (cid:1) ) in the lower half-plane.From this the authors derived a simpler evolution equation h∂ t z h ( t, E ) = i ( E − λ h ( s )) z h ( t, E ) + B h ( t, s, E ) ,s (cid:0) u h ( t, · ) (cid:1) ∼ (cid:90) (cid:12)(cid:12) z h ( t, E ) (cid:12)(cid:12) g ( E ) dE , (1.12)where B h ( t, x, E ) is a “driving term” derived from µ and e h . Then, usingan adiabatic approximation for the evolution of the nonlinear resonant statein term of instantaneous resonances and WKB expansions, an even furthersimplified differential equation for s (cid:0) u h ( t, · ) (cid:1) was obtained in the limit h → V = V ( t ). 7 .2 Adiabatic evolution of resonant states A rigorous study of the model (1.8) in the small h regime is a very vastprogram. In this connection, we remark that the lack of an error bound inthe adiabatic formulas used in [29] prevents to control the possible remainderterms in the asymptotic limit. Hence, a strong adiabatic theorem for resonantstates, with adiabatic parameter ε satisfyingln ε (cid:16) − /h , (1.13)seems to be a key point of this program. The adiabatic problem for res-onances can be considered following different approaches. One consists inusing the unitary propagator associated to the physical (selfadjoint) Hamil-tonian and study the adiabatic evolution of L states spectrally localizednear the resonant energy (or L functions obtained by truncating resonantstates). This point of view was adopted in [28] where the condition (1.13),connecting the adiabatic parameter to the resonance lifetime, was also takeninto account; in the case of a single time-dependent resonance E res ( t ) with:Im E res ( t ) ∼ Im E res (0) = e − ch an adiabatic formula was obtained on thespecific time range (Im E res (0)) − .A different approach consists in using a complex deformation to defineresonances as eigenvalues of a deformed (non-selfadjoint) Hamiltonian; in thisframework, the resonant states identify with L eigenvectors and the corre-sponding evolution problem is naturally formulated in terms of the deformedoperator. Then, an adiabatic approximation can be studied by adapting thestandard adiabatic theorem with gap condition to the non-selfadjoint case.(a similar strategy was implemented in [1]). It is worthwhile to remark thatthis requires uniform-in-time bounds for the deformed dynamical system (werefer to [27]): the lack of this condition, due to the complex deformation,is the main obstruction to implement such an approach in some relevantphysical models (including those we are considering here). In [20], the time-adiabatic evolution in Banach spaces has been considered, under a fixed gapcondition, for semigroups exhibiting an exponential growth in time. In thisframework, which could be adapted to the case of resonant states, the expo-nential growth of the dynamical system is compensated by the small error ofthe adiabatic approximation under analyticity-in-time assumptions and anadiabatic formula for the evolution of spectral projectors is provided with anerror which is small on a suitable range of time.More recently, an alternative approach to the adiabatic evolution of shaperesonances has been proposed [10]. For a 1D Schr¨odinger operator describ-ing the regime of quantum well in a semiclassical island, artificial (non-selfadjoint) interface conditions are added at the boundary of the potential’s8upport. Depending on the deformation parameter θ , these may be chosen insuch a way that the corresponding modified and complex deformed operatoris maximal accretive: hence, the deformed dynamical system allows uniform-in-time estimates and the adiabatic theory for isolated spectral sets can bedeveloped in the deformed setting. In particular, the authors show that arti-ficial interface conditions introduce small perturbations on shape resonances,preserving the relevant physical quantities (the exponentially small scales).Using an exterior complex deformation polynomially small w.r.t. the quan-tum scale h , an adiabatic theorem for the resonant states associated to shaperesonances is provided. In this framework, the adiabatic parameter ε = e − c/h can be fixed with any c > h -dependent deformation) is compensated by the smallerror of the adiabatic expansion which is now given by ε − δ where δ ∈ (0 , ε ,over time intervals of length ε − N for any fixed N ≥
0. In our framework, theevolution is no longer unitary and our first result says that we can arrange sothat the generator of our evolution has an imaginary part which is ≤ ε N forany N . A second result (for the moment limited to the case of one space di-mension), gives appropriate control on the resolvent in the same spaces , andwe get adiabatic approximation over long time intervals for exact solutionsof linear adiabatic evolution equations. (The multidimensional case will beattacked in a future work.) The aim of the present paper is to establish asymptotic approximations forsolutions of adiabatic evolution equations of the form( εD t + P ( t )) u ( t ) = 0 (1.14)that are valid over time intervals of length ε − N , with errors O ( ε N ) for ar-bitrary N >
0. Here P ( t ) = − h ∆ + V ( t, x ) is a self-adjoint Schr¨odingeroperator with a single well in a potential island which is assumed to generatea shape resonance with real energy E = E ( t ). Typically, ε should be compa-rable to the tunneling relaxation time for P ( t ) on a logarithmic scale. Moreprecisely, we should have ln(1 /ε ) (cid:16) /h . The approximations should be of9he form u ad ( t ) ∼ ( ν ( t ) + εν ( t ) + ... ) e − iε (cid:82) t λ ( s,ε ) ds (1.15)where ν j ( t ) are well-behaved smooth functions of t with values in the (byassumption) common domain of the P ( t ), and λ ( t, ε ) ∼ λ ( t ) + ελ ( t ) + ..., (1.16)where λ ( t ) is a shape resonance of P ( t ), satisfying0 ≤ −(cid:61) λ ( t ) ≤ e − o (1)) S ( t ) /h , (cid:60) λ ( t ) = E ( t ) + o ( t ) , (1.17)and ν ( t ) is a corresponding resonant state; ( P ( t ) − λ ( t )) ν ( t ) = 0. Here S ( t ) > P ( t ) − E ( t ) from the potentialwell to the sea surrounding the potential island.Since the non-linearity is concentrated to a bounded region, we can choosean ambient Hilbert function space H quite freely such that the space ofrestrictions of its elements to some neighborhood Ω of the island is equal to L (Ω).One such choice is H = L ( R n ). A nice feature with this choice is thatthe evolution (1.14) is norm preserving: (cid:107) u ( t ) (cid:107) H = (cid:107) u ( s ) (cid:107) H . A difficultywith this choice is that the resonant state ν ( t ) does not belong to L ( R n )(and λ ( t ) does not belong to the L -spectrum of P ( t )), so the adiabaticexpansion (1.15) can hold only locally in R n .Rather, we choose H to be a Hilbert space that contains the resonant state ν ( t ) and such that λ ( t ) belongs to the H -spectrum of P ( t ). When replacing L with some other Hilbert space we lose (in general) the self-adjointness ofthe operators P ( t ) and the corresponding unitarity of the evolution (1.14).The original non-linear problem is not time reversible, so we only wish tohave a good control of the solutions in the forward time direction. If we canchoose H , depending on ε but not on t , so that (cid:61) P ( t ) ≤ τ ( ε ) , (1.18)for P ( t ) as an unbounded operator H → H , where τ ( ε ) ≥
0, then if u ( t )solves (1.14) for t in some interval, we would have (cid:107) u ( t ) (cid:107) H ≤ e τ ( ε )( t − s ) (cid:107) u ( s ) (cid:107) H , for t ≥ s, (1.19)so the solution will grow at most exponentially with rate τ ( ε ) in the direc-tion of increasing time. Correspondingly, we can expect well-posedness forsolutions of the initial value problem, (cid:40) ( εD t + P ( t )) u ( t ) = 0 , ≤ t ≤ T,u (0) = u , (1.20)10ssuming, to fix the ideas, that P ( t ) is defined for t ∈ [0 , T ], T >
0. Then (cid:107) u ( t ) (cid:107) H ≤ e τ ( ε ) T (cid:107) u (cid:107) H and in order to avoid exponential growth of the upperbound we require τ ( ε ) T ≤ O (1) . With T = ε − N for some fixed N > τ ( ε ) ≤ O ( ε N ). As-sume that we can perform the adiabatic constructions in (1.15)–(1.17) with (cid:61) λ ( t, ε ) ≤ ε N +1 , N ≥ N , and that we have (1.18) with τ ( ε ) ≤ ε N +1 for0 ≤ t ≤ T , T ≤ ε − N . Then taking a suitable realization of u ad we expect tohave ( εD t + P ( t )) u ad = O ( ε N +1 ) in H , (cid:107) u ( t ) (cid:107) H = O (1) , for 0 ≤ t ≤ T and using (1.19) we would be able to solve( εD t + P ( t )) v = − ( εD t + P ( t )) u ad , (cid:107) v (0) (cid:107) H = 0 , with (cid:107) v (cid:107) = O ( ε N − N ). Then u ( t ) = u ad ( t ) + v ( t ) is an exact solution of( εD t + P ( t )) u = 0 on [0 , T ] with u − u ad = O ( ε N − N ) in H .The easiest choice of H , at first sight, would be to follow the methodof exterior complex distortions [2, 3, 32, 17] in the spirit of [35] so that H = L (Γ), where Γ ⊂ C n is a totally real manifold of real dimension n ,obtained as a deformation of R n , coinciding with R n along the island. Wedid non succeed with this particular choice however (see further commentsbelow). A. Faraj, A. Mantile and F. Nier [10] followed this path. Theydefined an operator (cid:101) P ( t ) from P ( t ) living on a distorted contour having anartificial interface condition between real part of the contour near the islandand the complex distorted part. This way the discrete spectrum of P ( t )needs no longer to consist of resonances of P ( t ), so the exact link with theoriginal evolution problem is disrupted.In this paper we use the spaces of [16]. Such spaces H = H (Λ υG ), 0 ≤ υ (cid:28) G ( x, ξ ) vanishing for large | ξ | , and very roughly H (Λ υG ) is the space of functions u ( x ) on R n , such that (cid:101) u ( x, ξ ) e − υG ( x,ξ ) /h ∈ L ( R n ), where (cid:101) u denotes a suitable FBI transform. Theassociated “I-Lagrangian” manifold Λ υG is given by (cid:61) ( x, ξ ) = υH G ( (cid:60) ( x, ξ )),where H G = ∂ ξ G · ∂ x − ∂ x G · ∂ ξ (cid:39) ( ∂ ξ G, − ∂ x G ) is the Hamilton field of G . Λ υG is then the natural classical phase space associated with H (Λ υG ).Thus if we consider a semi-classical Schr¨odinger operator P = − h ∆ + V ( x )with leading symbol p ( x, ξ ) = ξ + V ( x ) where V extends far enough inthe complex domain, the natural leading symbol of the unbounded operator P : H → H is p | Λ υG . Here, by Taylor expansion we have p | Λ υG ( x, ξ ) = p ( (cid:60) ( x, ξ )) + iυH G ( p )( (cid:60) ( x, ξ )) + O ( υ )= p ( (cid:60) ( x, ξ )) − iυH p ( G )( (cid:60) ( x, ξ )) + O ( υ ) , H p G ≥
0, then (at least up to O ( υ )) we have (cid:61) p | Λ υG ≤ P . In order todefine a resonance λ for P we need to choose G and υ so that near infinity (cid:61) p | Λ υG < (cid:61) λ when (cid:60) p | Λ υG belongs to a neighborhood of (cid:60) λ . This canbe obtained by choosing G to be an escape function, meaning roughly that H p G > / O (1) near infinity on p − ( E ) where E is some fixed real energyand we assume that (cid:60) λ ≈ E .The method of (small) contour distortions follows this scheme with thesymbol G ( x, ξ ) chosen to be linear or possibly affine linear in ξ . With suchrestrictions it is harder (maybe impossible) to find G so that H p G ≥ H p G > p − ( E ).For the construction of formal adiabatic solutions we will also need a goodcontrol over ( P ( t ) − z ) − for z on some small closed contour enclosing λ ( t ). The results of this paper concern the linear adiabatic theory for time depen-dent potentials with a well in an island, in a fairly general setting. We hopethat they will be useful for non-linear problems of the type described aboveand also that they are interesting in their own right. We study1) Semi-boundedness as in (1.18)2) Resolvent estimates in H (Λ υG )-spaces3) Adiabatic approximations over long time intervals.Here 3) will be a fairly direct consequence of 1) and 2), using general argu-ments from adiabatic constructions, that we shall review in Section 2, seealso Section 3.In Sections 4, 5 we review some of the theory in [16]. Let r ( x ) , R ( x ) bepositive smooth functions on R n satisfying (4.1): r ≥ , rR ≥ . Define (cid:101) r ( x, ξ ) = ( r ( x ) + ξ ) / as in (4.2) and the symbol spaces S ( m ) as in(4.3). We assume (4.4): m ∈ S ( m ) , r ∈ S ( r ) , R ∈ S ( R ) . Let 1 ≤ m ∈ S ( m ). We consider the formally self-adjoint semi-classicaldifferential operator in (4.6): P = (cid:88) | α |≤ N a α ( x ; h )( hD ) α , a α ∈ S ( m r −| α | ) , a α is a finite sum in powers of h as in (4.7) with leading term a α, ( x ).The full symbol of P for the standard left quantization will also be denotedby P (see (4.8)) and the semi-classical principal symbol will be denoted by p ( x, ξ ) ((4.13)): p ( x, ξ ) = (cid:88) | α |≤ N a α, ( x ) ξ α ∈ S ( m )where m = m ( x )( (cid:101) r ( x, ξ ) /r ( x )) N . We also have P ( x, ξ ; h ) ∈ S ( m ) (uni-formly with respect to h ). We make the ellipticity assumption (4.14), where p class is the classical (PDE) principal symbol in (4.12). Then for every fixedreal E the energy surface Σ E = p − ( E ) has the property that | ξ | ≤ O ( r ( x )) , for ( x, ξ ) ∈ Σ E . We say (see Definition 4.1) that the real-valued function G ∈ S ( (cid:101) rR ) is anescape function if H p G ( ρ ) ≥ m ( ρ ) O (1) on Σ E \ K, (1.21)for some compact set K . We make the technical assumption (4.22) (therestated with E replaced with 0, a reduction obtained by replacing p with p − E ): For every r > , there exists (cid:15) > , such that | p − E | ≥ (cid:15) m on R n \ (cid:91) ρ ∈ Σ B g ( ρ ) ( ρ, r ) . (1.22)Here g is the natural metric associated to the scales (cid:101) r , R , see (4.5).Proposition 4.2 (where again we took the case E = 0) states that ifwe have an escape function for a given energy E , then we can modify iton a bounded set to get an escape function G which vanishes on any givencompact set, such that H p G ≥ P = − h ∆ + V ( x ) , (1.23)where V is real-valued and ∂ αx V = o ( (cid:104) x (cid:105) −| α | ) , | x | → ∞ . (1.24)Then we take r ( x ) = 1, R ( x ) = (cid:104) x (cid:105) , m = ξ and P ( x, ξ ; h ) = p ( x, ξ ) = ξ + V ( x ). When E >
0, we have the escape function x · ξ and aftermultiplication with a cutoff χ ( p ( x, ξ ) − E ) we can also assume that G hascompact support in ξ . In this case (1.22) holds automatically.13et G = G ( x, ξ ) be real-valued and sufficiently small in S ( (cid:101) rR ). Let Λ G be the corresponding I-Lagrangian manifold (cid:61) ( x, ξ ) = H G ( (cid:60) ( x, ξ )), given in(5.1), which is also symplectic for (cid:60) σ , where σ = (cid:80) dξ j ∧ dx j is the complexsymplectic form on C n . We assume that | ξ | ≤ O ( r ( x )) on the support of G and define the weight function H on Λ G by (5.2). It is also of class S ( (cid:101) rR )when using the natural parametrization of Λ G in (5.1).Let T be an FBI-transform defined as in (5.3)–(5.6) so that T : E (cid:48) ( R n ) → C ∞ (Λ G ; C n +1 ). If m is an order function on Λ G ( m ∈ S ( m )), we define theHilbert spaces H (Λ G , m ) as in Definition 5.2 and put H (Λ G ) = H (Λ G , G = 0 this gives L ( R n ) with equivalence of norms. In Section 5.4 wereview pseudodifferential operators, Fourier integral operators and Toeplitzoperators, acting on these spaces.Let r , R , m ( x, ξ ) = m ( x )( (cid:101) r ( x, ξ ) /r ( x )) N be as above and let P be aformally self-adjoint h -differential operator as in (4.6), (4.7), fulfilling (4.14)as well as the technical assumption (4.22). We also make the exterior ana-lyticity assumption (4.11). If G ∈ S ( (cid:101) rR ) with | ξ | ≤ O ( r ( x )) on supp G , then(cf. (5.36)) we can view P as a bounded operator P : H (Λ υG , m ) → H (Λ υG ) , for 0 ≤ υ (cid:28)
1, provided that the coefficients a α,k of P are analytic in aneighborhood of the x -space projection of supp G . In Section 6 we prove afirst semiboundedness result: Theorem 1.1
Under the above assumptions, assume in addition that P hasan escape function G at energy E ∈ R . Let K ⊂ R n be compact, containingthe analytic singular support of P , i.e. the smallest closed set (cid:101) K such that thecoefficients of P (more precisely all the a α,k in (4.7) ) are analytic in R n \ (cid:101) K .Then we can find an escape function at energy E ; G ( x, ξ ; h ) ∼ G + hG + ... in S ( (cid:101) rR ) , supported in | ξ | ≤ O ( r ( x )) , where G = G near infinity on Σ E , π x supp G j , π x supp G are disjoint from a fixed neighborhood of K , such that for P as aclosed unbounded operator: H (Λ υG ) → H (Λ υG ) , we have (cid:61) ( P u | u ) H (Λ υG ) ≤ υ O ( h ∞ ) (cid:107) u (cid:107) H (Λ υG ,m / ) , (1.25) for υ ≥ and h > small enough. In the Schr¨odinger case ( m = (cid:104) ξ (cid:105) ), wecan replace H (Λ υG , m / ) with H (Λ υG ) . As we shall see, we can arrange so that (cid:107) u (cid:107) H (Λ υG ) = (cid:107) u (cid:107) L , when u ∈ L ( K ) . O ( h N ) forevery N > O ( e − N/h ) for every
N >
0. This improvement will beobtained with a scaling argument.Keeping the above assumptions, we also assume (7.1), (7.2): r = 1 , R ( x ) = (cid:104) x (cid:105) , m ( x ) = 1 , m = (cid:104) ξ (cid:105) N , as well as (7.3) which states that p ( x, ξ ) converges to a limiting polynomial p ∞ ( ξ ) when x → ∞ in the natural sense for the semi-norms of S ( m ). Themain fact that we exploit is now that in a region where x = µ (cid:101) x , | (cid:101) x | (cid:16) P ( x, hD x ; h ) can be viewed as an (cid:101) h -differential operator P ( µ (cid:101) x, (cid:101) hD (cid:101) x ; h ) with (cid:101) h = h/µ . Combination of this observation with Theorem 1.1 leads to (cf.(7.30)): Theorem 1.2
We make the assumptions of Theorem and the two ad-ditional assumptions above. Let π x : R n × R n (cid:51) ( x, ξ ) (cid:55)→ x ∈ R n . Thenuniformly for µ ∈ [1 , + ∞ [ , we can find an escape function G ( x, ξ, µ ; h ) ∼ G + hG + ... in S ( (cid:101) rR ) , with support in π − x (( R n \ B (0 , µ )) ∩ ( R nx × B (0 , r )) for some fixed r > andwith G = G on Σ E ∩ π − x ( R n \ B (0 , µ )) , independent of µ for | x | ≥ µ ,such that for P as an unbounded closed operator H (Λ υG ) → H (Λ υG ) , we have (cid:61) ( P u | u ) H (Λ υG ) ≤ υ O (( h/µ ) ∞ ) (cid:107) u (cid:107) H (Λ υG ,m / ) , ≤ υ (cid:28) . (1.26) In the Schr¨odinger operator case, we can replace H (Λ υG , m / ) in (1.26) with H (Λ υG ) . In this result we use a decoupling property which can be obtained with asuitable choice of norm in H (Λ υG ), namely that (cid:107) u (cid:107) H (Λ υG , = (cid:107) u (cid:107) L , when supp u ⊂ B (0 , µ/ . We next consider resolvent estimates. For simplicity we assume rightaway that P is a semi-classical Schr¨odinger operator as in (1.23), (1.24). Wewill also assume that we are in the 1-dimensional case; n = 1, even thoughwe now think that the higher dimensional case is within reach. With thehigher dimensional case in mind we will formulate certain statements as ifwe were in that general case, even though (for the moment) n = 1. In orderto fit with (1.24), we take r ( x ) = 1 , R ( x ) = (cid:104) x (cid:105) , m = (cid:104) ξ (cid:105) , P ( x, ξ ; h ) = p ( x, ξ ) = ξ + V ( x ). We keep the exterior analyt-icity assumption (4.11) which takes the form (8.3). Let E >
0. Let us firstconsider the non-trapping case (cf. Proposition 8.3).
Theorem 1.3
Assume that the H p -flow on Σ E = p − ( E ) is non-trapping (inthe sense that no maximal H p trajectory in p − ( E ) is contained in a compactset). Let G be as in Theorem , where we choose µ = h/(cid:15) where < (cid:15) (cid:28) h is a small parameter. Let ϑ > be small and fix υ > sufficiently small.If δ > , C > are respectively small and large enough, then for z in therange (8.45) : (cid:60) z ∈ [ E − δ / , E + δ / , − (cid:15) ϑ /C ≤ (cid:61) z ≤ /C, we have that P − z : H (Λ υ(cid:15)G , (cid:104) ξ (cid:105) ) → H (Λ υ(cid:15)G ) is bijective and m (cid:15) ( x ; h ) ( z − P ) − m (cid:15) ( x ; h ) = O (1) : H (Λ υ(cid:15)G ) → H (Λ υ(cid:15)G , (cid:104) ξ (cid:105) ) . Here we have put m (cid:15) ( x, ξ ) := h (cid:104) x (cid:105) ϑ + (cid:15) ϑ , (cid:15) ϑ = (cid:16) (cid:15)h (cid:17) ϑ (cid:15). We next consider a trapping case, namely that of a potential well in anisland. Let
E > (cid:98) R n be open (still with n = 1) and let U ⊂ ¨O be acompact subset. Assume (8.51), (8.52): V − E < R n \ ¨O , V − E > \ U , V − E ≤ U , diam d U = 0 , where d is the Lithner-Agmon distance associated to the metric( V − E ) + dx , ( V − E ) + = max( V − E, . Assume (8.53):The H p -flow has no trapped trajectories in p − ( E ) | R n \ ¨O . Let M ⊂ ¨O be a connected compact set with smooth boundary (i.e. acompact interval in the present 1D case) such that (8.54) holds: M ⊃ { x ∈ ¨O; d ( x, ∂ ¨O) ≥ (cid:15) } , for some small (cid:15) >
0. Let P denote the Dirichlet realization of P on M .16et J ( h ) ⊂ R be an interval tending to { E } as h →
0. Assume (8.67): P has no spectrum in ∂J ( h ) + [ − δ ( h ) , δ ( h )] , where the parameter δ ( h ) is small but not exponentially small;ln δ ( h ) ≥ − o (1) /h.σ ( P ) ∩ J ( h ) is a discrete set of the form { µ ( h ) , ..., µ m ( h ) } where m = m ( h ) = O ( h − n ) and we repeat the eigenvalues according to their multiplicity. LetΓ( h ) denote the set of resonances of P in J ( h ) − i [0 , (cid:15) ϑ /C ], C (cid:29)
1, alsorepeated according to their (algebraic) multiplicity. Assume (8.68): (cid:15) ≥ e − / ( Ch ) , for some C (cid:29)
1, so that (cid:16) (cid:15)h (cid:17) ϑ (cid:15) ≥ e O ( h ) − S h . Here S > U to ∂ ¨O. In Propo-sition 8.6 we recall a result from [16] when V is analytic everywhere and dueto Fujii´e, Lahmar-Benbernou, Martinez [11], for potentials that are merelysmooth on a bounded set, stating that there is a bijection b : { µ , ..., µ m } → Γ( h ) such that b ( µ ) − µ = (cid:101) O ( e − S /h ) := O ( e ω − S ) /h ) , ω = ω ( (cid:15) ) → , (cid:15) → . We give a proof in Section 8.
Theorem 1.4
For C (cid:29) sufficiently large, let z ∈ { z ∈ J ( h ) + i ] − (cid:15) ϑ /C, /C [; dist ( z, σ ( P ) ∩ J ( h )) = dist ( z, σ ( P )) } . (1.27) Assume either that m (the number of elements in Γ( h )) is equal to , or that dist ( z, σ ( P )) ≥ (cid:101) O ( e − S /h ) . Then we have (8.106) : ( z − P ) − = O ( h/δ ) + O (1) + O ( h/ dist ( z, Γ)) : m (cid:15) H sbd → m − (cid:15) D sbd , where the first two terms are holomorphic in the interior of the set (1.27) .Here H sbd = H (Λ υ(cid:15)G ) , D sbd = H (Λ υ(cid:15)G , (cid:104) ξ (cid:105) ) and υ , G are as in Theorem . We next turn to adiabatic expansions. Let I ⊂ R be an interval and let V t = V ( t, x ) ∈ C ∞ b ( I × R n ; R ) . (1.28)17ere C ∞ b (Ω) denotes the space of smooth functions on Ω that are boundedwith all their derivatives. We assume (cf. (8.3)) V t has a holomorphic extension (also denoted V t ) to { x ∈ C n ; |(cid:60) x | > C, |(cid:61) x | < |(cid:60) x | /C } such that V t ( x ) = o (1) , x → ∞ . (1.29) ∂ t V t ( x ) = 0 for | x | ≥ C, for some constant C > V ( t, x ) does not depend on h . However, when con-sidering a narrow potential well in an island, of diameter (cid:16) h , we will have tomake an exception and allow such an h -dependence in a small neighborhoodof the well.Let 0 < E − < E (cid:48)− < E (cid:48) + < E + < ∞ and let E ( t ) ∈ C ∞ b ( I ; [ E (cid:48)− , E (cid:48) + ]) . (1.31)We assume that V t − E ( t ) has a potential well in an island as above.Let ¨O = ¨O( t ) (cid:98) R n be a connected open set and let U ( t ) ⊂ ¨O( t ) becompact. Assume (cf. (8.51)), still with n = 1, V t − E ( t ) < R n \ ¨O( t ) ,> t ) \ U ( t ) , ≤ U ( t ) , (1.32)diam d t ( U ( t )) = 0 . (1.33)Here d t is the Lithner-Agmon distance on ¨O( t ), given by the metric ( V t − E ( t )) + dx .Also assume that with p t = ξ + V t ( x ),the H p t -flow has no trapped trajectories in p − t ( E ( t )) | R n \ ¨O( t ) . (1.34)It follows that d x V t (cid:54) = 0 on ∂ ¨O( t ) , (1.35)so ∂ ¨O( t ) is smooth and depends smoothly on t . Thus ¨O( t ) is a manifoldwith smooth boundary, depending smoothly on t . Further, U ( t ) dependscontinuously on t .For (cid:15) > M ( t ) = { x ∈ ¨O( t ); d t ( x, ∂ ¨O( t )) ≥ (cid:15) } , (1.36)18o M ( t ) (cid:98) ¨O( t ) is a compact set with smooth boundary, depending smoothlyon t . (Here we use the structure of d t ( x, ¨O( t )) that follows from (1.35),see Sections 9, 10 of [16].) More precisely, for every fixed t (consequentlysuppressed from the notation) the function φ ( x ) = d ( x, ∂ ¨O) in ¨O is analyticin the interior, continuous up to the boundary, where it vanishes, and solvesthe eikonal equation |∇ φ | = V ( x ). Over a neighborhood of the boundary,we have the Lagrangian manifold Λ, defined as the flow out of ∂ ¨O × { ξ = 0 } under the Hamilton flow of q = ξ − V ( x ). The manifold Λ has a simplefold over the boundary and Λ φ : ξ = φ (cid:48) ( x ) describes “one of the two coveringhalves” of Λ. It is quite well known then that if we choose analytic localcoordinates y = ( y , ..., y n − , V ) = ( y (cid:48) , V ) near a boundary point, then φ is ananalytic function of y (cid:48) , V / and has a convergent expansion φ = a ( y ) V / + a ( y ) V + ... with a k analytic and a > I in (1.28) to be a very long interval, weintroduce the following uniformity assumption:( V t , E ( t )) ∈ K , ∀ t ∈ I, where K is a compact subset of { V ∈ C ∞ b ( R n ; R ); V satisfies (1.29) with a fixed constant C } × [ E (cid:48)− , E (cid:48) + ]such that ( V, E ) satisfies the assumptions (1.30), with a fixed C as well as (1.32), (1.33), (1.34). (1.37)Let P ( t ) denote the Dirichlet realization of P ( t ) = − h ∆ + V t ( x ) on M ( t ). If we enumerate the eigenvalues of P ( t ) in ] E − , E + [ in increasing order(repeated with multiplicities) we know (as a general fact for 1-parameterfamilies of self-adjoint operators), that they are uniformly Lipschitz functionsof t . Let µ ( t ) = µ ( t ; h ) be such an eigenvalue and assume, µ ( t ; h ) = E ( t ) + o (1) , h → , uniformly in t. (1.38) µ ( t ; h ) is a simple eigenvalue and σ ( P ( t )) ∩ [ E ( t ) − δ ( h ) , E ( t ) + δ ( h )] = { µ ( t ; h ) } . (1.39)Here, as above, δ ( h ) > δ ( h ) ≥ − o (1) /h, h → . (1.40)In addition to (8.68), we assume (9.57), so we have (9.114): e − / ( C h ) ≤ (cid:15) ≤ min( h/C , δ ) , C , C (cid:29) . Let λ ( t ) be the unique resonance of P ( t ) in D ( µ ( t ) , δ ( h )) (the open discin C with center µ ( t ) and radius δ ( h )), so that λ ( t ) − µ ( t ) = (cid:101) O ( e − S ( t ) /h ).19s we shall see in (9.60): ∂ kt λ ( t ) = O ( δ ( h ) − k ) , k ≥ . Let e ( t ) be the corresponding resonant state, ( P ( t ) − λ ( t )) e ( t ) = 0,uniquely determined up to a factor ± t ) bythe condition (cid:90) R n e ( t, x ) dx = 1 . As we shall see in Section 9, we can find an escape function G as in Theorem1.2 which applies simultaneously to all ( P ( t ) , E ( t )). Moreover, we can choose G such that G ( x, − ξ ) = − G ( x, ξ ) and with this choice the bilinear scalarproduct (cid:104) u | v (cid:105) = (cid:90) u ( x ) v ( x ) dx is well-defined and bounded on H (Λ υ(cid:15)G ) × H (Λ υ(cid:15)G ). Then e = O (1) in H (Λ υ(cid:15)G ). Recall that H sbd = H (Λ υ(cid:15)G ), D sbd = H (Λ υ(cid:15)G , (cid:104) ξ (cid:105) ) where υ > ∂ kt e ( t ) = O (1) (cid:18) h(cid:15) ϑ (cid:19) k in D sbd for k ≥ . Using the resolvent estimates in Theorem 1.4 we will establish (as Proposition9.2) the following result:
Theorem 1.5
Under the assumptions above there exist two formal asymp-totic series, ν ( t, ε ) ∼ ν ( t ) + εν ( t ) + ε ν ( t ) + ... in C ∞ ( I ; D sbd ) , (1.41) λ ( t, ε ) ∼ λ ( t ) + ελ ( t ) + ε λ ( t ) + ... in C ∞ ( I ) , (1.42) such that ( εD t + P ( t ) − λ ( t, ε )) ν ( t, ε ) ∼ as a formal asymptotic series in C ∞ ( I ; H sbd ) . Here, ∂ kt ν j = O (1)( h/ ˆ (cid:15) ϑ ) j + k in D sbd , j ≥ , k ≥ , (1.44) ∂ kt λ j = O (1)( h/ ˆ (cid:15) ϑ ) j − k , j ≥ , k ≥ , (1.45) where ˆ (cid:15) ϑ := (cid:15) ϑ max(1 , (cid:15) ϑ / ( δh )) / = min( (cid:15) ϑ , ( (cid:15) ϑ δh ) / ) . (1.46)20e continue the discussion under the assumptions of Theorem 1.5. Putfor N ≥ ν ( N ) = ν + εν + ... + ε N ν N , (1.47) λ ( N ) = λ + ελ + ... + ε N λ N , N ≥ . (1.48)Then by the proof of Theorem 1.5 (cf. (2.4)),( εD t + P ( t ) − λ ( N ) ) ν ( N ) = r ( N +1) , (1.49)where r ( N +1) = ε N +1 D t ν N − (cid:88) j,k ≤ Nj + k ≥ N +1 ε j + k λ j ν k . (1.50)In the following, we assume that ε h ˆ (cid:15) ϑ (cid:28) . (1.51)Recall from (1.40) that δ = δ ( h ) is small, but not exponentially small andthat (cid:15) ϑ = ( (cid:15)/h ) ϑ (cid:15) . Then (1.51) holds if we assume that ε is exponentiallysmall: 0 < ε ≤ O (1) exp ( − / ( Ch )) , for some C > , (1.52)and choose (cid:15) ≥ ε ϑ ) − α , (1.53)for some α ∈ ]0 , / (4(1 + ϑ ))[. We also assume (9.95), stating that (cid:15) ≤O ( ε /N ) for some N > ∂ kt r ( N +1) = O (1) ε (cid:32) ε h ˆ (cid:15) ϑ (cid:33) N +1 (cid:18) h ˆ (cid:15) ϑ (cid:19) k in D sbd . (1.54)Also, (cid:107) ν ( N ) ( t ) (cid:107) H sbd = (1 + O ( εh / ˆ (cid:15) ϑ )) (cid:107) ν ( t ) (cid:107) H sbd (cid:16) . (1.55)From (1.25) with µ in (8.26), µ = h/(cid:15) , we get: (cid:61) ( P ( t ) u | u ) H sbd ≤ O ( (cid:15) ∞ ) (cid:107) u (cid:107) H sbd . (1.56)Let I (cid:51) t (cid:55)→ u ( t ) ∈ H (Λ υ(cid:15)G , (cid:104) ξ (cid:105) ) be continuous such that ∂ t u is continuouswith values in H (Λ υ(cid:15)G , (cid:104) ξ (cid:105) − ). Assume that u is a solution of( εD t + P ( t )) u ( t ) = 0 .
21t then follows from (1.56) that (cid:107) u ( t ) (cid:107) H sbd ≤ e O ( (cid:15) ∞ )( t − s ) (cid:107) u ( s ) (cid:107) H sbd , t ≥ s. (1.57)From (1.56) we will derive a resolvent bound which leads to the fact thatfor every u ∈ D sbd and every s ∈ I , there exist u ∈ C ( I ∩ [ s, ∞ [; D sbd ) ∩ C ( I ∩ [ s, ∞ [; H sbd ) such that( εD t + P ( t )) u ( t ) = 0 for s ≤ t ∈ I, u ( s ) = u . (1.58)Again the solution satisfies (1.57). When P ( t ) = P ( t ) is independent of t , this follows from the Hille–Yosida theorem. In the general case we canuse [22, Theorem 6.1 and Remark 6.2].This allows us to define the forward fundamental matrix E ( t, s ), I (cid:51) t ≥ s ∈ I of εD t + P ( t ): (cid:40) ( εD t + P ( t )) E ( t, s ) = 0 , t ≥ s,E ( t, t ) = 1and we have (cid:107) E ( t, s ) (cid:107) L ( H sbd , H sbd ) ≤ exp(( t − s ) O ( (cid:15) ∞ )) , t ≥ s, t, s ∈ I. (1.59)If v ∈ C ( I ; H sbd ) vanishes for t near inf I , we can solve ( εD t + P ( t )) u = v on I by u ( t ) = iε (cid:90) t inf I E ( t, s ) v ( s ) ds. Now return to (1.47)–(1.49) with λ j , ν j as in Theorem 1.5 and r ( N +1) satisfying (1.54). We notice that λ ( N ) = λ + O (1) ε ε h ˆ (cid:15) ϑ . (1.60)We can choose ν ( t ) = e ( t ) implying that λ = 0 and (1.60) improves to λ ( N ) = λ + O (1) ε (cid:32) ε h ˆ (cid:15) ϑ (cid:33) . (1.61)See Remark 9.3. We choose ν = e in the remainder of this introduction.Assume, to fix the ideas, that 0 ∈ I , and restrict the attention to I + = { t ∈ I ; t ≥ } . From (1.49), we get( εD t + P ( t )) u ( N ) = ρ ( N +1) , t ∈ I + , (1.62)22here u ( N ) = e − i (cid:82) t λ ( N ) ds/ε ν ( N ) , ρ ( N +1) = e − i (cid:82) t λ ( N ) ds/ε r ( N +1) . (1.63)By (1.55), (1.54), we have (cid:107) ρ ( N +1) (cid:107) H sbd = O (1) ε (cid:32) ε h ˆ (cid:15) ϑ (cid:33) N +1 (cid:107) u ( N ) (cid:107) H sbd . (1.64)Using again (1.56), we get as in Section 9, (cid:107) u ( N ) ( t ) (cid:107) ≤ e O (1) tε − / ( ε / h/ ˆ (cid:15) ϑ ) N +1 (cid:107) u ( N ) (0) (cid:107) , t ∈ I + . (1.65)Assume (9.108): (sup I ) ε − (cid:32) ε h ˆ (cid:15) ϑ (cid:33) N +1 ≤ O (1) . (1.66)Then, for t ∈ I + , (cid:107) u ( N ) ( t ) (cid:107) H sbd ≤ O (1) , (cid:107) ρ ( N +1) ( t ) (cid:107) H sbd ≤ O (1) ε (cid:32) ε h ˆ (cid:15) ϑ (cid:33) N +1 . (1.67)Using the fundamental matrix E to correct the error ρ ( N +1) we have theexact solution u = u ( N )exact , u = u ( N ) − iε (cid:90) t E ( t, s ) ρ ( N +1) ( s ) ds (1.68)of the equation ( εD t + P ( t )) u = 0 on I + . From (1.66) we get sup I ≤ ε − N , (1.69)for some fixed finite N . Then by (1.59) (cid:107) E ( t, s ) (cid:107) L ( H sbd , H sbd ) ≤ e O ( ε ∞ ) = 1 + O ( ε ∞ ) , (1.70)and using this and (1.67) in (1.68), we get (cid:107) u − u ( N ) (cid:107) H sbd ≤ O (1) ε − (sup I ) ε (cid:32) ε h ˆ (cid:15) ϑ (cid:33) N +1 . (1.71)This estimate is the main result of the present work. Let us recollect theassumptions and the general context in the following theorem (same as The-orem 9.5 below). 23 heorem 1.6 Let V t = V ( t, x ) ∈ C ∞ b ( I × R n ; R ) , where n = 1 , < E −
Let H be a separable complex Hilbert space, let I ⊂ R be a compact intervaland let P = P ( t ) : H → H be a closed densely defined operator, depending24n t ∈ I such that(H1) The domain D = D ( t ) is independent of t ∈ I and the domain norms (cid:107) u (cid:107) D ( t ) are uniformly equivalent to each other in the sense that thereexists a constant C ≥ C − (cid:107) u (cid:107) D ( s ) ≤ (cid:107) u (cid:107) D ( t ) ≤ C (cid:107) u (cid:107) D ( s ) , s, t ∈ I , u ∈ D (H2) P ( t ) ∈ C ∞ ( I ; L ( D , H )) in the natural sense: the successive derivativesare uniformly bounded D → H and again differentialble in the uniformsense.In this section we shall also assume(H3) P ( t ) has a simple eigenvalue λ ( t ) which depends continuously on t andis isolated from the rest of the spectrum:dist ( λ ( t ) , σ ( P ( t )) \ { λ ( t ) } ) ≥ /C, where C > t .It follows from these assumptions that λ ( t ) is a smooth function of t . In thefollowing result we review the formal adiabatic construction. Proposition 2.1
There exist two asymptotic series ν ( t, ε ) ∼ ν ( t ) + εν ( t ) + ε ν ( t ) + ... in C ∞ ( I ; D ) , (2.1) λ ( t, ε ) = λ ( t ) + ελ ( t ) + ... in C ∞ ( I ) , (2.2) where ν ( t ) is non-vanishing, such that ( εD t + P ( t ) − λ ( t, ε )) ν ( t, ε ) ∼ , (2.3) as an asymptotic series in C ∞ ( I ; H ) . Proof.
We insert the developments for ν and λ into (2.3) and try to cancelthe successive powers of ε :( εD t + P ( t ) − λ ( t, ε )) ν ( t, ε ) =( P ( t ) − λ ( t )) ν ( t ) + ε (( D t − λ ( t )) ν ( t ) + ( P − λ ) ν ( t ))+ ... + ε k (( P ( t ) − λ ( t )) ν k + ( D t − λ ) ν k − − λ ν k − − ... − λ k ν )+ ... (2.4)25n order to annihilate the ε term it is necessary and sufficient to let ν ( t ) bea non-vanishing eigenvector associated to λ ( t ), which depends smoothly on t and we fix such a choice.By assumption, ν ( t ) is unique up to a smooth non-vanishing scalar factor.The corresponding spectral projection (independent of the scalar factor) is π ( t ) = 12 πi (cid:90) γ ( t ) ( z − P ( t )) − dz, (2.5)where γ ( t ) is the oriented boundary of the disc D ( λ ( t ) , r ) and r > π ( t ) is a projection of rank 1 andhence of the form π ( t ) u = ( u | δ ( t )) ν ( t ) . (2.6)This projection and its adjoint, π ( t ) ∗ v = ( v | ν ( t )) δ ( t ) depend smoothlyon t and we deduce that δ ( t ) (like ν ( t )) depends smoothly on t . Since π ( t )( P ( t ) − λ ( t )) = 0, we have (( P ( t ) − λ ( t )) u | δ ( t )) = 0 for all u ∈ D and it follows that δ ( t ) ∈ D ( P ( t ) ∗ ) and that ( P ( t ) ∗ − λ ( t )) δ ( t ) = 0. From π ( t ) = π ( t ) it follows that ( ν ( t ) | δ ( t )) = 1.By holomorphic functional calculus, we know that H = R (1 − π ( t )) ⊕R ( π ( t )) = N ( π ( t )) ⊕R ( π ( t )) = δ ( t ) ⊥ ⊕ C ν ( t ) . (2.7)Further, P ( t ) : δ ( t ) ⊥ → δ ( t ) ⊥ is a closed densely defined operator withspectrum equal to σ ( P ( t )) \{ λ ( t ) } . From this we conclude that the equation( P − λ ( t )) u = v (2.8)has a solution precisely when v ⊥ δ ( t ) and when this condition is fulfilledthe general solution is of the form u = (cid:101) u + zν ( t ), where (cid:101) u is the uniquesolution in δ ( t ) ⊥ and z ∈ C is arbitrary.In order to annihilate the ε term in (2.4) we need to find ν ( t ) ∈ D depending smoothly on t such that( P ( t ) − λ ( t )) ν ( t ) = λ ( t ) ν ( t ) − D t ν ( t ) . (2.9)As we have just seen, this equation can be solved precisely when0 = ( λ ( t ) ν ( t ) − D t ν ( t ) | δ ( t )) = λ ( t ) − ( D t ν | ν ) , (2.10)so we choose λ = ( D t ν | ν ). Choose a smooth solution ν ( t ) of (2.9) (whichis unique up to a term z ( t ) ν ( t ) where z ( t ) is a smooth scalar function).Then, to annihilate the ε term, we have the equation,( P ( t ) − λ ( t )) ν ( t ) + ( D t − λ ( t )) ν ( t ) − λ ν = 0 , and we see that the solvabilty with respect to ν ( t ) imposes a unique choiceof λ . By iterating this argument we get the proposition. (cid:50) emark 2.2 (a) Let λ, ν be as in the proposition and let θ ( t, ε ) ∼ θ ( t ) + εθ ( t ) + ... in C ∞ ( I ) . Then, (cid:101) λ = λ + ε∂ t θ, (cid:101) ν = e iθ ν is another pair as in the Proposition. Indeed, ∼ e iθ ( εD t + P − λ ) e − iθ (cid:101) ν = ( εD t + P − (cid:101) λ ) (cid:101) ν. Now, any function (cid:101) λ ∼ λ ( t ) + ε (cid:101) λ ( t ) + ... in C ∞ ( I ) is of the form (cid:101) λ = λ + ε∂ t θ for a suitable θ as above (which is unique up to a constant C ( ε ) ∼ C + εC + ... ) and we conclude that λ ( t, ε ) in the proposition can be anyasymptotic series as in (2.2) , with leading term λ given in (H3) .(b) If ( λ, ν ) and ( λ, (cid:101) ν ) are two pairs as in the proposition, then there exists C ( ε ) ∼ C + εC + ... , such that (cid:101) ν = C ( ε ) ν . In fact, writing (cid:101) ν = C ( t ) ν and the corresponding equation for (cid:101) ν ; ( P − λ ) (cid:101) ν = C ( λ − D t ) ν − ( D t C ) ν , we see that D t C has to vanish, so that C is constant. Repeat the argumentfor ( εD t + P − λ ) (cid:18) (cid:101) ν − C νε (cid:19) = 0 , to see that (cid:101) ν = C ν + εC ν + O ( ε ) , where C is a constant. By iterationwe get the statement. Let P ( t ) satisfy the assumptions of Section 2 except for the assumption (H3)that we generalize to(H4) For some integer N ≥ P ( t ) has a group of N eigenvalues λ ( t ) , ... , λ N ( t ) counted with their multiplicities, depending continuously on t and isolated from the rest of the spectrum:dist ( { λ · ( t ) } , σ ( P ( t )) \ { λ · ( t ) } ) ≥ /C, where C > t . Here { λ · ( t ) } = { λ j ( t ); 1 ≤ j ≤ N } .27his assumption can be reformulated as follows: • There exists a simple closed C loop, γ = γ t : S → C , enclosing somenon-empty part of the spectrum, such that dist ( γ t ( S ) , ∩ σ ( P ( t ))) ≥ /C and such that the spectral projection π ( t ) = 12 πi (cid:90) γ t ( z − P ( t )) − dz (3.1)has finite rank, necessarily equal to some constant N ≥ U ( t, ε ) ∼ U ( t ) + εU ( t ) + ... ∈ C ∞ ( I ; L ( C N , D )) (3.2)and Λ( t, ε ) ∼ Λ ( t ) + ε Λ ( t ) + ... ∈ C ∞ ( I ; L ( C N , C N )) , (3.3)such that ( εD t + P ( t )) U ( t, ε ) − U ( t, ε )Λ( t, ε ) = 0 (3.4)as a formal powerseries with values in L ( C N , H ), σ (Λ ( t )) = { λ · ( t ) } , (3.5) U ( t ) is injective and U ( t ) = π ( t ) U ( t ) . (3.6)As in the case of a single eigenvalue ( N = 1) we substitute (3.2), (3.3)into (3.4) and try to annihilate the successive powers of ε . This leads to theequations, P U − U Λ = 0 , (3.7) P U − U Λ + D t U − U Λ = 0 , (3.8)... P U k − U k Λ + D t U k − − U k − Λ − U k − Λ ... − U Λ k = 0 , (3.9)... As for (3.7), we let U ( t ) : C N → R ( π ( t )) (3.10)be any injective map which depends smothly on t and then take Λ ( t ) = U ( t ) − P ( t ) U ( t ), where U − denotes the inverse of (3.10). Then σ (Λ ( t )) = σ (cid:16) P ( t ) |R ( π ( t )) (cid:17) ,
28o (3.5) is fulfilled.In order to solve (3.8) we first choose Λ ( t ) so that π ( D t U − U Λ ) = 0 , (3.11)i.e. Λ = U − π D t U , where U − denotes the inverse of the operator in (3.10).Then we look for U with (1 − π ) U = U , (3.12)i.e. U : C N → R (1 − π ) ∩ D , and it suffices to find such a (smooth familyof) map(s) such that P U − U Λ + (1 − π ) D t U = 0 . (3.13)Here P is identified with P |R (1 − π ) : R (1 − π ) → R (1 − π ), which is closed,densely defined and satisfies σ (cid:16) P |R (1 − π ) (cid:17) ∩ σ (Λ ) = ∅ . (3.14) Lemma 3.1
Let H be a complex separable Hilbert space and let A : H → H be closed and densely defined. Let B ∈ L ( C N , C N ) and assume that σ ( A ) ∩ σ ( B ) = ∅ . (3.15) Then for every V ∈ L ( C N , H ) , there is a unique U ∈ L ( C N , D ( A )) suchthat AU − U B = V. (3.16) Proof.
Decompose C N = (cid:77) λ ∈ σ ( B ) E λ , where E λ is the spectral subspace, so that B : E λ → E λ and B | E λ = λ + N ,where N = N λ is nilpotent. It suffices to find, for every λ ∈ σ ( B ), a uniquelinear operator U = U λ : E λ → D ( A ) such that AU − U ( λ + N ) = V λ , where V λ = V | E λ . We write this as ( A − λ ) U − U N = V λ (3.17)and notice that when N = 0, the unique solution is U = ( A − λ ) − V λ .29n the general case we look for U of the form U = ( A − λ ) − (cid:101) U , (cid:101) U ∈L ( E λ , H ) and (3.17) becomes (cid:101) U − (cid:101) N ( (cid:101) U ) = V λ , (3.18)where (cid:101) N ( (cid:101) U ) := ( A − λ ) − (cid:101) U N.
It then suffices to observe that (cid:101) N is nilpotent, so that (3.18) has a uniquesolution. (cid:50) By a simple Neumann series argument, if A = A ( t ), B = B ( t ), V = V ( t )depend smoothly on a real parameter and D ( A ( t )) is independent of t , thenthe same holds for U ( t ).Applying Lemma 3.1 and the above observation to (3.13), we get a uniquesolution U ( t ) : C N → D ∩ R (1 − π ) which is smooth in t . Thus, there is aunique solution ( U , Λ ) to (3.11)–(3.13).Assuming that U , ..., U k − , Λ , ..., Λ k − have been constructed, we solve(3.9) in the same way: First, make the unique choice of Λ k for which π ( D t U k − − U k − Λ − U k − Λ ... − U Λ k ) = 0 . (3.19)Then, let U k be the unique map: C N → R (1 − π ) ∩ D , such that P U k − U k Λ + (1 − π ) ( D t U k − − Λ U k − ... − Λ k U ) = 0 . Summing up the discussion, we have
Proposition 3.2
The problem (3.2) – (3.6) has a solution with U k ∈ C ∞ ( I ; L ( C N , D )) , Λ k ∈ C ∞ ( I ; L ( C N , C N )) . The solution is unique if we first choose U , Λ as in (3.7) and then requirethat (1 − π ( t )) U k ( t ) = U k ( t ) for k ≥ . We keep the assumption of the preceding section. Recall the notion of adia-batic spectral projections, [26, 27] in the presentation of [34]. Consider P ( t, εD t ; ε ) = εD t + P ( t ) (3.20)as a vector valued ε -pseudodifferential operator (see e.g. [9]) with symbol P ( t, τ ; ε ) = τ + P ( t ) . (3.21)30hen for z ∈ neigh ( γ t ), where γ t is as in (3.1), we define the formal resolvent( z − P ) − = S ( t, εD t , z ; ε ) , (3.22)as a formal ε -pseudodifferential operator with symbol S ( t, τ, z ; ε ) ∼ S ( t, τ, z ) + εS ( t, τ, z ) + ..., (3.23)defined for t ∈ I , z − τ ∈ neigh ( γ t ) and obtained by the standard ellipticparametrix construction, so that S ( t, τ, z ) = ( z − τ − P ( t )) − . The corre-sponding adiabatic projection is the formal ε -pseudodifferential operator, π ( t, εD t ; ε ) = 12 πi (cid:90) γ t ( z − P ) − dz, (3.24)defined on the symbol level, for τ ∈ neigh (0 , C ).Using the property, S ( t, τ, z ; ε ) = S ( t, τ − z, ε )it is shown in [34] that the symbol π ( t, τ ; ε ) is independent of τ , so that π ( t, εD t ; ε ) u = π ( t, ε ) u ( t ) , (3.25)where π ( t ; ε ) = π ( t ) + επ ( t ) + ... ∈ C ∞ ( I ; L ( H , D )) . (3.26) π ( t ) is the spectral projection for P ( t ) in (3.1). Moreover (cf. (16), (17) in[34]), π ( t ; ε ) = π ( t ; ε ) , (3.27)[ εD t + P ( t ) , π ( t ; ε )] = 0 . (3.28) Proposition 3.3
Let U ∼ U ( t ) + εU ( t ) + ... , Λ ∼ Λ ( t ) + ε Λ ( t ) + ... be asolution of the problem (3.2) – (3.4) . Put (cid:101) U = π ( t, ε ) U . Then (cid:101) U ∼ U + ε (cid:101) U + ... (with (cid:101) U = π U ), and ( (cid:101) U , (cid:101)
Λ) := ( (cid:101)
U , Λ) is a solution of (3.2) – (3.4) and wehave π ( t, ε ) (cid:101) U ( t, ε ) = (cid:101) U ( t, ε ) . (3.29) In view of (3.29) and the fact that π is asymptotically a projection, we shallsay that R ( (cid:101) U ) ⊂ R ( π ) (pointwise in t ). roof. By (3.4), (3.28), we get( εD t + P ( t )) (cid:101) U − (cid:101) U Λ = π (( εD t + P ( t ) U − U Λ) = 0 (cid:50)
Proposition 3.4
Let ( U, Λ) , ( (cid:101) U , (cid:101) Λ) be two solutions of the problem (3.2) – (3.4) , with R ( U ) , R ( (cid:101) U ) ⊂ R ( π ) in the sense of (3.29) . Assume that U satisfies (3.6) . Then ∃ ! M ( t ; ε ) ∼ M ( t ) + εM ( t ) + ... ∈ C ∞ ( I ; L ( C N , C N )) (3.30) such that (cid:101) U = U M. (3.31)
Conversely, if U solves (3.2) - (3.6) and M is of the form (3.30) with M ( t ) invertible, then ( (cid:101) U , (cid:101) Λ) solves (3.2) – (3.5) where (cid:101) U := U M, (cid:101)
Λ := M − εD t ( M ) + M − Λ M. (3.32) Proof.
We first prove the converse part by direct calculation( εD t + P ( t )) (cid:101) U = (( εD t + P ( t )) U ) M + U εD t ( M )= U (Λ M + εD t ( M ))= U M ( M − Λ M + M − εD t ( M ))= (cid:101) U (cid:101) Λ . We next prove the direct part. Let ( U, Λ), ( (cid:101)
U , (cid:101)
Λ) be as in the beginningof the proposition, both solving (3.2)–(3.4) with πU = U , π (cid:101) U = (cid:101) U and suchthat U satisfies (3.6).Writing (cid:101) U = (cid:101) U + ε (cid:101) U + ... , we conclude that (cid:101) U maps C N → R ( π )pointwise in t . U ( t ) : C N → R ( π ( t )) has the same property and is bijective.Hence there is a unique M ( t ) : C N → C N , smooth in t , such that (cid:101) U ( t ) = U ( t ) M ( t ). From the proof of the “converse” part, we see that V ( t ) := (cid:101) U ( t ) − U ( t ) M ( t ) ∼ : εV , ( t ) + ε V , ( t ) + ... solves (3.2)–(3.4) with Λ replaced by a new matrix ε Λ ( t ; ε ). We also have πV = 0, so π ( t ) V , ( t ) = 0 and hence ∃ M ( t ) : C N → C N , such that V , ( t ) = U ( t ) M ( t ). Then V ( t ) := (cid:101) U ( t ) − U ( t )( M ( t ) + εM ( t )) ∼ : ε V , ( t ) + ε V , ( t ) + ... and π ( t ) V ( t ) = 0, so π ( t ) V , ( t ) = 0. Iterating this procedure we get M ( t ) ∼ M ( t ) + εM ( t ) + ... with the required properties. (cid:50) .3 Extension to the case of variable ε Consider the evolution equation( D s + P ( s )) ν ( s ) = 0 (3.33)on some large interval I , where P ( s ) are closed densely defined operatorswith common domain D . Assume that ∂ ks P ( s ) = O ( ε ( s ) k ) , k = 0 , , , ..., (3.34)as bounded operators from D to H . Here the function ε ( s ) is assumed tosatisfy ε > , ∂ ks ε = O ( ε k +1 ) , k ≥ . (3.35) Remark 3.5
Under the same assumptions, if (cid:101) ε ≥ ε is a second function on I which satisfies (3.35), then (3.34) holds with ε replaced by (cid:101) ε . Let f ( s ) be the strictly increasing function, uniquely determined up to aconstant, by f (cid:48) ( s ) = ε ( s ) . (3.36)Then, if t = f ( s ), we have D s = f (cid:48) ( s ) D t = ε ( s ) D t and (3.33) takes the form,( ε ( g ( t )) D t + P ( g ( t ))) u ( t ) = 0 , u ( t ) = ν ( g ( t )) . (3.37)Here, g := f − .Differentiating f ( g ( t )) = t , we first get f (cid:48) ( g ( t )) g (cid:48) ( t ) = 1, so g (cid:48) ( t ) = 1 ε ( g ( t )) . (3.38)Differentiating m times, where m ≥
2, we get f (cid:48) ( g ( t )) ∂ mt g ( t ) + m (cid:88) k =2 (cid:88) mj ≥ ,m ... + mk = m C m ,...,m k f ( k ) ( g ( t )) ∂ m t g ( t ) · ... · ∂ m k t g ( t ) = 0 . Assuming by induction, that ∂ (cid:101) mt g = O (1 /ε ( g ( t ))) , (cid:101) m < m, we get ε ( g ( t )) ∂ mt g + m (cid:88) k =2 (cid:88) m + .. + m k = m O (1) ε ( g ( t )) k ε ( g ( t )) − k = 0 ,
33o we get ∂ mt g = O (1 /ε ( g ( t ))) , m ≥ . (3.39)Now, ∂ mt ( P ( g ( t ))) = m (cid:88) k =1 C m ,..,m k P ( k ) ( g ( t )) ∂ m t g...∂ m k t g = (cid:88) O (1) ε k − k = O (1) . (3.40)Similarly, ∂ mt ε ( g ( t )) = (cid:88) O (1) ε ( k ) ∂ m t g...∂ m k t g = O ( ε ( g ( t ))) . (3.41)This shows that (3.37) is a very nice semi-classical equation. This section is a review of some material from [16, 14] and we add someremarks for later use. We adopt the frame work of [16]: Choose two positivesmooth scale functions r ( x ), R ( x ) on R n with r ≥ , rR ≥ , (4.1)Let (cid:101) r ( x, ξ ) = ( r ( x ) + ξ ) ∈ C ∞ ( R n ) . (4.2)If 0 < m ∈ C ∞ ( R n ) we say that a ∈ C ∞ ( R n ) belongs to the space S ( m ), if | ∂ αx ∂ βξ a | ≤ C α,β m ( x, ξ ) R ( x ) −| α | (cid:101) r ( x, ξ ) −| β | . (4.3)We will always assume that the weight m and the scale functions belong totheir own symbol classes: m ∈ S ( m ) , r ∈ S ( r ) , R ∈ S ( R ) . (4.4)It follows that (cid:101) r ∈ S ( (cid:101) r ). The naturally associated metric on R n in thespirit of H¨ormander’s Weyl calculus of pseudodifferential operators [18] isgiven by g = (cid:18) dξ (cid:101) r (cid:19) + (cid:18) dxR (cid:19) . (4.5)It is slowly varying, but another important assumption of that calculus willnot be satisfied in our case however, namely the σ -temperance.34ften, a and even m will depend on the semi-classical parameter h , and itwill then be implicitly assumed that all estimates involved in the statements a ∈ S ( m ) and m ∈ S ( m ) are uniform with respect to h (and possibly otherparameters as well). We define h k S ( m ) := S ( h k m ). When a : R n → R n is a smooth map and g is a smooth metric on R n , we say that a is of class S ( m ) for the metric g , if g a ( x,ξ ) ( ∂ αx ∂ βξ a ( x, ξ )) satisfy the estimates in (4.3)uniformly.Let 1 ≤ m ( x ) ∈ S ( m ) and let P = P ( x, hD ; h ) be a semi-classicaldifferential operator on R n of the form P = (cid:88) | α |≤ N a α ( x ; h )( hD x ) α , (4.6)where a α ( x ; h ) ∈ S ( m ( x ) r −| α | ) and a α ( x ; h ) = N −| α | (cid:88) k =0 h k a α,k ( x ) , a α,k ∈ S ( m r −| α | ( rR ) − k ) . (4.7)Such operators form an algebra in the natural way. Then h k a α,k ξ α ∈ S (cid:0) m ( (cid:101) r/r ) N ( h/ ( (cid:101) rR )) k (cid:1) and we have the full semi-classical symbol for the standard left quantization P ( x, ξ ; h ) = (cid:88) | α |≤ N a α ( x ; h ) ξ α ∈ S ( m ) , where m ( x, ξ ) = m ( x )( (cid:101) r/r ) N (4.8)We can write P ( x, ξ ; h ) = p ( x, ξ ) + hp ( x, ξ ) + h p ( x, ξ ) + ... + h N p N ( x, ξ ) , (4.9)where p j ∈ S ( m ( (cid:101) rR ) − j ) , (4.10)We also assume analyticity in x near infinity: ∃ C > P extends to a holomorphic functionin { x ∈ C n ; |(cid:60) x | > C, |(cid:61) x | ≤ R ( (cid:60) x ) /C } and thesymbol properties above extend in the natural sense. (4.11)This could be formulated more directly in terms of the coefficients a α in (4.6).We assume that P is formally self-adjoint, so that the classical and thesemi-classical principal symbols, given respectively by p class ( x, ξ ) = (cid:88) | α | = N a α, ( x ) ξ α (4.12)35nd p ( x, ξ ) = (cid:88) | α |≤ N a α, ( x ) ξ α (4.13)are real-valued. We make the ellipticity assumption in the classical PDEsense, p class ( x, ξ ) ≥ m ( x )( | ξ | /r ) N , ( x, ξ ) ∈ R n . (4.14)This implies for the zero energy surface,Σ = { ( x, ξ ) ∈ R n ; p ( x, ξ ) = 0 } , (4.15)that | ξ | ≤ Const . r ( x ) on Σ . (4.16)The same holds on Σ E := p − ( E ) for every fixed E , but we shall mainlyconcentrate on the case E = 0 to simplify the notation (observing that afterreplacing p with p − E , we are reduced to that case).We are particularly interested in the following situation: P = − h ∆ + V ( x ) − P ( x, ξ ; h ) = p ( x, ξ ) = ξ + V ( x ) − , (4.18)We will assume that V is smooth, real-valued and extends holomorphicallyto the set in (4.11) and tends to 0 when x → ∞ in that set. This enters intothe general framework with r = 1 , R = (cid:104) x (cid:105) , m ( x ) = 1 , m = (cid:104) ξ (cid:105) . (4.19)We next discuss escape functions. If a j ∈ S ( m j ), j = 1 ,
2, then a a ∈ S ( m m ) and H a a = { a , a } ∈ S (cid:16) m m (cid:101) rR (cid:17) . (4.20)(4.20) remains valid if we weaken the assumption on a , a to a j ∈ ˙ S ( m j ) for j = 1 , S ( m ) denote the space of smooth functions a on R n which satisfy the estimates (4.3) for all non-vanishing ( α, β ) ∈ N n . Definition 4.1
A real-valued function G ∈ ˙ S ( (cid:101) rR ) is called an escape func-tion if there exists a constant C and a compact set K ⊂ R n such that H p G ( ρ ) ≥ m ( ρ ) C , for all ρ ∈ Σ \ K. (4.21)36hen specifying the energy level we say that G in the definition above is anescape function at energy 0. More generally we can define escape functionsfor p at a real energy E , replacing Σ by Σ E .As we have already noticed, | ξ | ≤ r ( x ) on Σ , so m (cid:16) m there. Also, H p G ∈ S ( m ) for all G ∈ ˙ S ( (cid:101) rR ), so when G is an escape function, we have H p G (cid:16) m on Σ near infinity.We will also need to know that | p | cannot be very small away from Σ and therefore make the following assumption:For every r > , there exists (cid:15) > , such that | p | ≥ (cid:15) m on R n \ (cid:91) ρ ∈ Σ B g ( ρ ) ( ρ, r ) . (4.22)Here, g is the metric in (4.5) and B g ( ρ ) ( ρ, r ) denotes the open ball withcenter ρ and radius r for the constant metric g ( ρ ).For (cid:15) > [ − (cid:15) ,(cid:15) ] = { ρ ∈ R n ; | p ( ρ ) | ≤ (cid:15) } . (4.23)The assumption (4.22) implies that for every r >
0, there exists (cid:15) > [ − (cid:15) ,(cid:15) ] ⊂ (cid:91) ρ ∈ Σ B g ( ρ ) ( ρ, r ) . (4.24)From (4.21), (4.24) and the fact that H p G ∈ S ( m ) it follows that there exist C , (cid:15) > K ⊂ R n such that H p G ( ρ ) ≥ m ( ρ ) C , for all ρ ∈ Σ [ − (cid:15) ,(cid:15) ] \ K. (4.25)For the Hamilton field, H G = ∂ ξ G ( x, ξ ) · ∂ x − ∂ x G ( x, ξ ) · ∂ ξ , we get when G ∈ ˙ S ( (cid:101) rR ), (cid:107) H G (cid:107) g (cid:16) | ∂ ξ G | R + | ∂ x G | (cid:101) r = O (1) . (4.26)Using that p ∈ S ( m ), we also have H p G = (cid:101) rp (cid:48) ξ · G (cid:48) x (cid:101) r − Rp (cid:48) x · G (cid:48) ξ R = O ( m ) (cid:107) H G (cid:107) g . Thus, if G is an escape function we get with (cid:15) , K as in (4.25), (cid:107) H G (cid:107) g (cid:16) [ − (cid:15) ,(cid:15) ] \ K (4.27)37nd here, H p G ≥ ( m/C ) (cid:107) H G (cid:107) g . Also notice that (cid:107) H p (cid:107) g = O (cid:16) m (cid:101) rR (cid:17) . (4.28)Until further notice we restrict the attention to Σ [ − (cid:15) ,(cid:15) ] . We next review theappendix in [14] and especially how to improve the escape function G bymodifying it on a bounded set. Let] − τ − ( ρ ) , τ + ( ρ )[ (cid:51) t (cid:55)→ exp tH p ( ρ )be the maximal H p -integral curve through the point ρ ∈ Σ [ − (cid:15) ,(cid:15) ] , where0 < τ ± ( ρ ) ≤ + ∞ are lower semi-continuous.If K ⊂ Σ [ − (cid:15) ,(cid:15) ] is a compact subset as in (4.25), then there exists a finitenumber T = T ( K ) > − T ( K ) < G < T ( K ) on K. (4.29)The set { ρ ∈ Σ [ − (cid:15) ,(cid:15) ] ; G ( ρ ) ≥ T ( K ) } is invariant under the H p -flow in thepositive time direction: G ( ρ ) ≥ T ( K ) = ⇒ G (exp tH p ( ρ )) ≥ T ( K ) , ≤ t < τ + ( ρ ) . If G ( ρ ) ≥ T ( K ), (cid:15) > tH p ( ρ ) ∈ B g ( ρ ) ( ρ, (cid:15) ) for 0 ≤ t ≤ t ( (cid:15) ) (cid:101) rR/m , when t ( (cid:15) ) > G (exp tH p ( ρ )) willincrease by (cid:16) (cid:101) rR ( ρ ) ≥ ρ replaced by exp t ( (cid:15) ) H p ( ρ ) and so on. The tra-jectory will have to go through infinitely many balls as above and we con-clude that G (exp tH p ( ρ )) → + ∞ when τ → τ + ( ρ ), for every ρ ∈ Σ [ − (cid:15) ,(cid:15) ] ∩ G − ([ T ( K ) , + ∞ [). Similarly, G (exp tH p ( ρ )) → −∞ when 0 ≥ t → − τ − ( ρ )for every ρ ∈ G − (] − ∞ , − T ( K )]).By a similar argument, if K ⊂ Σ [ − (cid:15) ,(cid:15) ] is a sufficiently large compactset containing K , then for every ρ ∈ G − ([ − T ( K ) , T ( K )]) \ K , we haveexp tH p ( ρ ) (cid:54)∈ K , t ∈ R , and G (exp tH p ( ρ )) → ±∞ when t → ± τ ± ( ρ ).Define the outgoing and incoming tails Γ + , Γ − ⊂ Σ [ − (cid:15) ,(cid:15) ] respectively, byΓ ± = { ρ ∈ Σ [ − (cid:15) ,(cid:15) ] ; exp tH p ( ρ ) (cid:54)→ ∞ , t → ∓ τ ∓ ( ρ ) } . (4.30)In [14] it was shown that Γ ± are closed sets,Γ + ⊂ G − (] − T ( K ) , + ∞ [) , Γ − ⊂ G − (] − ∞ , T ( K )[) , and that Γ + ∩ G − (] − ∞ , T ]) , Γ − ∩ G − ([ − T, + ∞ [)38re compact for every T ∈ R . In particular the trapped set Γ + ∩ Γ − is acompact subset of G − (] − T ( K ) , T ( K )[) andΓ ± = { ρ ∈ Σ [ − (cid:15) ,(cid:15) ] ; exp tH p ( ρ ) → Γ + ∩ Γ − , t → ∓ τ ∓ ( ρ ) } . (4.31)Having fixed T = T ( K ) above, let (cid:101) K ⊂ G − (] − T ( K ) , T ( K )[) be acompact set containing the trapped set Γ + ∩ Γ − . For ρ ∈ G − ( T ( K )), define σ + ( ρ ) = sup { t ∈ ]0 , τ − ( ρ )[; exp( − [0 , t ] H p )( ρ ) ⊂ G − (] − T ( K ) , T ( K )[) \ (cid:101) K } . (4.32)When ρ is outside the set K above, assumed to be large enough, the ( − H p )-trajectory through ρ will hit G − ( − T ( K )) without reaching (cid:101) K or get trappedand σ + ( ρ ) is the corresponding hitting time which depends locally smoothlyon ρ . For ρ ∈ K ∩ G − ( T ( K )) it may also happen that the trajectory hits (cid:101) K at the finite time σ + ( ρ ) or converges to Γ + ∩ Γ − without hitting (cid:101) K , inwhich case σ + ( ρ ) = τ − ( ρ ) = + ∞ .Notice that σ + is a lower semi-continuous function. Define the opensubset Ω + of G − ( T ( K )) × [0 , + ∞ [ byΩ + := { ( ρ, t ) ∈ G − ( T ( K )) × [0 , + ∞ [; 0 ≤ t < σ + ( ρ ) } . (4.33)Then (cid:101) Ω + = { exp( − tH p )( ρ ); ( ρ, t ) ∈ Ω + } is an open subset of G − (] − T ( K ) , T ( K )]) and the mapΩ + (cid:51) ( ρ, t ) (cid:55)→ exp( − tH p )( ρ ) ∈ (cid:101) Ω + (4.34)is a diffeomorphism. We have (cid:101) Ω + = G − (] − T ( K ) , T ( K )]) \ Γ − ( (cid:101) K ) , (4.35)where Γ − ( (cid:101) K ) denotes the incoming (cid:101) K -tail , defined asΓ − ( (cid:101) K ) = Γ − ∪ { exp( − tH p )( ρ ); ρ ∈ (cid:101) K, ≤ t < τ − ( ρ ) } . (4.36)The intersection of Γ − ( (cid:101) K ) with G − ([ − T, + ∞ [) is compact for every T ∈ R .Let f + ∈ C ∞ ( (cid:101) Ω + ; ]0 , + ∞ [) be equal to H p G near G − ( T ( K )) and outsidesome bounded set. Define G + ∈ C ∞ ( (cid:101) Ω + ) by H p G + = f + , G + = G on G − ( T ( K )) (4.37)39nd observe first that G + = G near G − ( T ( K )). By choosing f + large enoughwe may arrange so thatlim sup (cid:101) Ω + (cid:51) ν → ∂ (cid:101) Ω + G + ( ν ) ≤ − T ( K ) − C , C > , lim sup (cid:101) Ω + (cid:51) ν → G − ( − T ( K )) G + ( ν ) ≤ − T ( K ) , (4.38)where ∂ (cid:101) Ω + denotes the boundary of (cid:101) Ω + as a subset G − (] − T ( K ) , T ( K )]),so that ∂ (cid:101) Ω + ⊂ Γ − ( (cid:101) K ). Since G + = G near G − ( T ( K )), we can extend G + by G to a smooth function G + ∈ C ∞ ( (cid:101) Ω + ∪ G − (] T ( K ) , ∞ [)).By construction, if χ ∈ C ∞ ( R ) and supp χ ⊂ [ − T ( K ) , + ∞ [, then χ ◦ G + is well-defined in C ∞ ( R n ).Next we briefly introduce the analogous quantities, Ω − , G − : For ρ ∈ G − ( − T ( K )), define σ − ( ρ ) = sup { t ∈ ]0 , τ + ( ρ )[; exp([0 , t ] H p )( ρ ) ⊂ G − (] − T ( K ) , T ( K )[) \ (cid:101) K } . (4.39) σ − is a lower semi-continuous function. Let us also define the open subsetΩ − of G − ( − T ( K )) × [0 , + ∞ [ byΩ − := { ( ρ, t ) ∈ G − ( − T ( K )) × [0 , + ∞ [; 0 ≤ t < σ − ( ρ ) } . (4.40)Then (cid:101) Ω − := { exp( tH p )( ρ ); ( ρ, t ) ∈ Ω − } is an open subset of G − ([ − T ( K ) , T ( K )[) and the mapΩ − (cid:51) ( ρ, t ) (cid:55)→ exp( tH p )( ρ ) ∈ (cid:101) Ω − (4.41)is a diffeomorphism. We have (cid:101) Ω − = G − ([ − T ( K ) , T ( K )[) \ Γ + ( (cid:101) K ) , (4.42)where Γ + ( (cid:101) K ) denotes the outgoing (cid:101) K -tail , defined asΓ + ( (cid:101) K ) = Γ + ∪ { exp( tH p )( ρ ); ρ ∈ (cid:101) K, ≤ t < τ + ( ρ ) } . (4.43)The intersection of Γ + ( (cid:101) K ) with G − (] − ∞ , T ]) is compact for every T ∈ R .Let f − ∈ C ∞ ( (cid:101) Ω + ; ]0 , + ∞ [) be equal to H p G near G − ( − T ( K )) and out-side some bounded set. Define G − ∈ C ∞ ( (cid:101) Ω − ) by H p G − = f − , G − = G on G − ( − T ( K )) (4.44)40nd observe that G − = G near G − ( − T ( K )). By choosing f − large enoughwe may arrange so thatlim inf (cid:101) Ω − (cid:51) ν → ∂ (cid:101) Ω − G − ( ν ) ≥ T ( K ) + 1 C , C > , lim inf (cid:101) Ω − (cid:51) ν → G − ( T ( K )) G − ( ν ) ≥ T ( K ) , (4.45)where ∂ (cid:101) Ω − denotes the boundary of (cid:101) Ω + as a subset G − ([ − T ( K ) , T ( K )[), sothat ∂ (cid:101) Ω − ⊂ Γ + ( (cid:101) K ).Since G − = G near G − ( − T ( K )), we can extend G − by G to a smoothfunction G − ∈ C ∞ ( (cid:101) Ω − ∪ G − (] − ∞ , − T ( K )[)). By construction, if χ ∈ C ∞ ( R ) and supp χ ⊂ ] − ∞ , T ( K )], then χ ◦ G − is well-defined in C ∞ ( R n ).With T = T ( (cid:101) K ), we define (cid:101) G = χ + ◦ G + + χ − ◦ G − ∈ C ∞ (Σ [ − (cid:15) ,(cid:15) ] ) , (4.46)where • χ ± ∈ C ∞ ( R ; R ), ± χ ± ≥ • χ + ( t ) + χ − ( t ) = t , • χ (cid:48) + > − T, + ∞ [, • χ (cid:48)− > −∞ , T [, • supp χ + = [ − T, + ∞ [, supp χ − =] − ∞ , T ],We notice that (cid:101) G = G outside a bounded subset of G − ([ − T ( K ) , T ( K )])and that (cid:101) G − (0) ⊃ { ρ ; G + ( ρ ) ≤ − T ( K ) , G − ( ρ ) ≥ T ( K ) } ⊃ Γ + ( (cid:101) K ) ∩ Γ − ( (cid:101) K ) . (4.47)It is also clear that (cid:101) G − (0) ⊂ G − (] − T, T [). Moreover, we can choose f + , f − so thatthe set G + ( ρ ) ≤ − T ( K ) is contained in an arbitrarily small neighbor-hood of G − ( − T ( K )) ∪ Γ − ( (cid:101) K ),the set G − ( ρ ) ≥ T ( K ) is contained in an arbitrarily small neighborhoodof G − ( T ( K )) ∪ Γ + ( (cid:101) K ). 41utside a bounded set, we have (cid:101) G = G , (cid:107) H G (cid:107) g (cid:16) H p G ≥ m/ O (1)by (4.27), (4.21). In a bounded set, we use (4.46) and get H p (cid:101) G = ( χ (cid:48) + ◦ G + ) f + + ( χ (cid:48)− ◦ G − ) f − (cid:16) χ (cid:48) + ◦ G + + χ (cid:48)− ◦ G − , (4.48) (cid:107) H (cid:101) G (cid:107) g = ( χ (cid:48) + ◦ G + ) (cid:107) H G + (cid:107) g + ( χ (cid:48)− ◦ G − ) (cid:107) H G − (cid:107) g (cid:16) χ (cid:48) + ◦ G + + χ (cid:48)− ◦ G − , (4.49)so H p (cid:101) G (cid:16) m (cid:107) H (cid:101) G (cid:107) g uniformly on Σ [ − (cid:15) ,(cid:15) ] , (4.50)in addition to the fact that H p (cid:101) G ∈ S ( m ) and (cid:107) H (cid:101) G (cid:107) g (cid:16) H p (cid:101) G > + ( (cid:101) K ) ∩ Γ − ( (cid:101) K ).We now strengthen the assumption G ∈ ˙ S ( (cid:101) rR ) to G ∈ S ( (cid:101) rR ) . (4.51)Then (cid:101) G ∈ S ( (cid:101) rR ) and outside a bounded set we have (cid:101) G = O ( (cid:101) rR ) (cid:107) H (cid:101) G (cid:107) g . Choose χ ± so that χ ± = O ( χ (cid:48)± ) uniformly on any bounded set.Then from (4.46), (4.49), we conclude that (cid:101) G = O ( (cid:101) rR ) (cid:107) H (cid:101) G (cid:107) g , uniformly on Σ [ − (cid:15) ,(cid:15) ] . (4.52)Assume m (cid:16) [ − (cid:15) ,(cid:15) ] . (4.53)Let χ ∈ C ∞ (] − (cid:15) , (cid:15) [; [0 , − (cid:15) / , (cid:15) /
2] and define globally, G ( ρ ) = χ ( p ( ρ )) (cid:101) G ( ρ ) ∈ C ∞ ( R n ) , (4.54)with the convention that G = 0 outside Σ [ − (cid:15) ,(cid:15) ] . By (4.53) we have χ ( p ( ρ )) ∈ S (1) and hence G ∈ S ( (cid:101) rR ) by (4.51) and the subsequent observation. Then H p G = χ ( p ) H p (cid:101) G, (4.55) H G = χ ( p ) H (cid:101) G + (cid:101) Gχ (cid:48) ( p ) H p . (4.56)42rom the last equation, (4.28) and (4.53), we get (cid:107) H G (cid:107) g ≤ χ ( p ) (cid:107) H (cid:101) G (cid:107) g + O (1) | χ (cid:48) ( p ) || (cid:101) G | m (cid:101) rR ≤ χ ( p ) (cid:107) H (cid:101) G (cid:107) g + O (1) | χ (cid:48) ( p ) (cid:101) G | (cid:101) rR = O (1) , leading to (cid:107) H G (cid:107) g ≤ O (1) (cid:32) χ ( p ) (cid:107) H (cid:101) G (cid:107) g + χ ( p ) | (cid:101) G | ( (cid:101) rR ) (cid:33) , in view of the standard estimate, χ (cid:48) = O ( χ / ) for non-negative smoothfunctions. Now apply (4.52) to get (cid:107) H G (cid:107) g ≤ O (1) χ ( p ) (cid:107) H (cid:101) G (cid:107) g ≤ O (1) χ ( p ) (cid:107) H (cid:101) G (cid:107) g , (4.57)where the last inequality follows from (4.26) which also holds for (cid:101) G . Com-bining this with (4.50), (4.55), we get H p G ≥ m O (1) (cid:107) H G (cid:107) g . (4.58)We sum up the constructions in Proposition 4.2
Let r, R, (cid:101) r be as in (4.1) – (4.4) , define the metric g by (4.5) .Let P , p m be as in (4.6) – (4.10) , where ≤ m ∈ S ( m ) . Assume (4.14) with p class as in (4.12) . Define the energy slice Σ [ − (cid:15) ,(cid:15) ] by (4.23) for some (cid:15) > and let G ∈ S ( (cid:101) rR ) be an escape function in the sense of Definition and assume (4.22) , so that (4.25) holds if (cid:15) > is small enough, and fixsuch a choice of (cid:15) . Let (cid:101) K ⊂ Σ [ − (cid:15) ,(cid:15) ] be a compact set which contains thetrapped set Γ + ∩ Γ − (cf. (4.30) ). Define the outgoing and incoming (cid:101) K -tails Γ + ( (cid:101) K ) , Γ − ( (cid:101) K ) by (4.43) , (4.36) , so that (cid:98)(cid:101) K := Γ + ( (cid:101) K ) ∩ Γ − ( (cid:101) K ) ⊂ Σ [ − (cid:15) ,(cid:15) ] isa compact set; “the H p -convex hull” of (cid:101) K .Then, after modifying G on a bounded set to a new function (cid:101) G , we canachieve that • H p (cid:101) G (cid:16) m (cid:107) H (cid:101) G (cid:107) g uniformly on Σ (cid:15) , • H p (cid:101) G > outside any fixed given neighborhood of (cid:98)(cid:101) K , • (cid:101) G = 0 in a neighborhood of (cid:98)(cid:101) K .If we also assume (4.53) and define G ∈ S ( (cid:101) rR ) as in (4.54) , then wehave (4.58) . Microlocal approach to resonances ([16])
In this section we review some basic notions developed in the first half of[16].
Let G ∈ ˙ S ( (cid:101) rR ) be real-valued. Then the manifoldΛ G = { ( x, ξ ) ∈ C n ; (cid:61) ( x, ξ ) = H G ( (cid:60) ( x, ξ )) } (5.1)is I-Lagrangian, i.e. Lagrangian in C n for the real symplectic form −(cid:61) σ ,where σ = (cid:80) dξ j ∧ dx j is the complex symplectic form. Since Λ G is I-Lagrangian, d ( −(cid:61) ( ξ · dx ) | Λ G ) = −(cid:61) σ | Λ G = 0 and since Λ G is topologicallytrivial, we know that −(cid:61) ( ξ · dx ) | Λ G is exact and hence = dH for some smoothfunction H ∈ C ∞ (Λ G ). The primitive H is unique up to a constant and wecan choose H = −(cid:60) ξ · (cid:61) x + G ( (cid:60) ( x, ξ )) = G ( (cid:60) ( x, ξ )) − (cid:60) ξ · G (cid:48) ξ ( (cid:60) ( x, ξ )) . (5.2)If we also assume that G is small in ˙ S ( (cid:101) rR ), then Λ G is R -symplectic, i.e.a symplectic submanifold of C n , equipped with the symplectic form (cid:60) σ . Inother words, σ | Λ G is a (real) symplectic form on Λ G and we have the volumeelement dα = 1 n ! σ n | Λ G . (5.1) gives a parametrization R n (cid:51) ρ (cid:55)→ ρ + iH G ( ρ ) of Λ G and we can thendefine symbol spaces S ( m ) = S ( m, Λ G ) of functions on Λ G by pulling backfunctions and weights to R n . In particular, we define the scales (cid:101) r and R bythis pull back. Let λ = λ ( α ) ∈ S ( (cid:101) rR − , Λ G ) be positive, elliptic (in the sensethat λ is non-vanishing and 1 /λ ∈ S (( (cid:101) rR − ) − , Λ G )) and put φ ( α, y ) = ( α x − y ) α ξ + iλ ( α )( α x − y ) , α = ( α x , α ξ ) ∈ Λ G , y ∈ C n . (5.3)This will be the phase in our FBI-transform.The amplitude will be a C n +1 -valued smooth function t ( α, y ; h ) on Λ G × C ny which is affine linear in y . When discussing symbol properties of suchfunctions we restrict the attention to a region | y − α x | < O (1) R ( α x ) , (5.4)44nd with this convention, we require that t ∈ h − n/ S ( (cid:101) r n/ R − n/ ) and that t , ∂ y t , ..., ∂ y n t are maximally linearly independent in the sense that with t treated as a column vector, (cid:12)(cid:12) det (cid:0) t ∂ y t ... ∂ y n t (cid:1)(cid:12)(cid:12) (cid:16) R − n (cid:16) h − n (cid:101) r n R − n (cid:17) n +1 . (5.5)Notice that the determinant is independent of y . If e , e , ..., e n is the canon-ical basis in C n +1 , we can choose t ( α, y ) = t ( α ; h ) + n (cid:88) ( α x j − y j ) t j ( α ; h ) , where, t j = t j e j , and t j ( α ; h ) = t ( α ; h ) R for j > t ∈ h − n/ S ( (cid:101) r n/ R − n/ ) is elliptic. Remark 5.1 If s ( α, y ; h ) is a second amplitude with the same properties as t , then it is not hard to show that there exists U ( α ; h ) : C n +1 → C n +1 ,independent of y and invertible, such that U, U − ∈ S (1) and s ( α, y ; h ) = U ( α ; h ) t ( α, y ; h ) . Let χ ∈ C ∞ ( B (0 , /C )) be equal to one in B (0 , / (2 C )), where C > T : D (cid:48) ( R n ) → C ∞ (Λ G ; C n +1 ) by T u ( α ; h ) = (cid:90) e ih φ ( α,y ) t ( α, y ; h ) χ α ( y ) u ( y ) dy, (5.6)where χ α ( y ) = χ (( y −(cid:60) α x ) /R ( (cid:60) α x )). Here the domain of integration is equalto R n and the integral is defined as the bilinear scalar product of u ∈ D (cid:48) ( R n )and a test function in C ∞ ( R n ).We assume from now on that G belongs to S ( (cid:101) rR ). We also assume: ∃ g = g ( x ) ∈ S ( rR ) , such that G ( x, ξ ) − g ( x )has its support in a region where | ξ | ≤ O ( r ( x ))and G ( x, ξ ) − g ( x ) is sufficiently small in S ( rR ) . (5.7)We will also consider the more special situation, when G ∈ S ( m G ): ∃ g = g ( x ) ∈ S ( m G ) , such that G ( x, ξ ) − g ( x )has its support in a region where | ξ | ≤ O ( r ( x )) . and G ( x, ξ ) − g ( x ) is sufficiently small in S ( m G ) . (5.8)45ere m G ≤ (cid:101) rR is an order function, and we have put m G ( x ) = m G ( x, H be given in (5.2). Then H ∈ S ( (cid:101) rR ). Under the more restrictiveassumption (5.8), we have H ∈ S ( m G ) . (5.9)Using T we shall define the function spaces H (Λ G , m ), essentially byrequiring that T u ∈ L (Λ G , m e − H/h dα ) =: L (Λ G , m ) . Here, m is an order function: 0 < m ∈ S ( m ). An intuitive reason for theappearance of H here is the following: The function f ( y, θ ) = −(cid:61) y · θ + G ( (cid:60) y, θ ) , θ ∈ R n (5.10)is a nondegenerate phase function on C n × R n in the sense of H¨ormander’stheory of Fourier integral operators (apart from a homogeneity condition)with θ as the fiber variables. The corresponding critical manifold C f is givenby f (cid:48) θ ( y, θ ) = 0 : (cid:61) y = G (cid:48) η ( (cid:60) y, θ )and the associated I-Lagrangian manifold is { ( y, i ∂ y f ( y, θ )); ( y, θ ) ∈ C f } = Λ G . We are beyond the scope of H¨ormander’s theory, but from this it is naturalto define the space H (Λ G , m ) by saying that a distribution u should belongto it when T u ∈ L (Λ G , m e − (cid:101) H/h dα ), where (cid:101) H ( α ) = v . c . ( y,θ ) ( −(cid:61) φ ( α, y ) + f ( y, θ )) . Here v . c . ( y,θ ) indicates that we take the critical value with respect to thevariables ( y, θ ). The critical point is nondegenerate and given by ( y, θ ) =( α x , (cid:60) α ξ ) and we get (cid:101) H ( α ) = H ( α ) . Letting Λ = R n , we can find an FBI-transform T : D (cid:48) ( R n ) → C ∞ (Λ ; C n +1 )given by T u ( β ; h ) = (cid:90) e ih φ ( β,y ) s ( β, y ; h ) (cid:101) χ β ( y ) u ( y ) dy, T in the sense of (5.13) below, provided that φ ( β, y ) = ( β x − y ) · β ξ + iλ ( β )( β x − y ) and s , (cid:101) χ β are chosen suitably. First we need a bijection Λ G (cid:51) α (cid:55)→ β ∈ Λ and we define β = β ( α ) by imposing the condition { β } = Λ ∩ { ( y, − ∂ y φ ( α, y )); y ∈ C n } , which gives β = ( β x , β ξ ) = (cid:18) (cid:60) α x + (cid:61) α ξ λ ( α ) , (cid:60) α ξ − λ ( α ) (cid:61) α x (cid:19) . (5.11)This gives a bijection Λ G → Λ with inverse β → α ( β ), both having thenatural symbol properties. We define the elliptic element 0 < λ ∈ S ( (cid:101) rR − )by λ ( β ) = λ ( α ( β )) . (5.12)By construction the two quadratic polynomials φ ( α, · ) and φ ( β, · ) have thesame gradients and Hessians at the point y = β x , so they differ by a constant(independent of y ). More explicitly, φ ( α, y ) = φ ( α, β x ) + φ ( β, y ) . Finally, choose s ( β, y ) = t ( α, y ) , (cid:101) χ β ( y ) = χ α ( y ) . Then
T u ( α ; h ) = e ih φ ( α,β x ) T u ( β ; h ) , (5.13)which expresses the equivalence of T and T .It follows that if we identify order functions on Λ G and on Λ in a naturalway, then we have the equivalence T u ∈ L (Λ G , m e − H/h dα ) ⇔ T u ∈ L (Λ , m e − F/h dβ ) , where F = H + (cid:61) φ ( α, β x ) = v . c . y,θ ( −(cid:61) φ ( β, y ) + f ( y, θ )) . (5.14)Let G , G ∈ S ( (cid:101) rR ) be as above and let f , f and F , F be the corre-sponding functions. In [16] it was shown, using (5.14) and a correspondinginverse “Legendre” formula, that we have the equivalence, G ≤ G ⇔ F ≤ F . (5.15)From this and the description with the help of T it will follow that we havethe inclusion H (Λ G , m ) ⊂ H (Λ G , m ), when G ≤ G .47 .3 Sobolev spaces with exponential phase space wei-ghts Let G satisfy (5.7) and be sufficiently small in S ( (cid:101) rR ). Define H as in (5.2),let m be an order function on Λ G and let T be an associated FBI-transformas in (5.6). In [16] it is shown that T is injective on C ∞ ( R n ) and also onmore general Sobolev spaces with exponential weights, by the constructionof an approximate left inverse of T which works with exponentially smallerrors. Definition 5.2 H (Λ G , m ) is the completion of C ∞ ( R n ) for the norm (cid:107) u (cid:107) H (Λ G ,m ) = (cid:107) T u (cid:107) L (Λ G ,m e − H/h dα ) . (5.16)The following facts were established in [16]: • H (Λ G , m ) is a Hilbert space • If we modify the choice of λ and t in the definition of T , we get thesame space H (Λ G , m ) and the new norm is uniformly equivalent to theearlier one, when h → • When G = g ( x ) is independent of ξ and m = m ( x ), we get H (Λ G , m ) = L ( R n ; m e − g ( x ) /h dx )with uniform equivalence of norms. More generally, when m ( x, ξ ) = m ( x )( (cid:101) r ( x, ξ ) /r ( x )) N , N ∈ R , then H (Λ G , m ) is the naturally definedexponentially weighted Sobolev space. Remark 5.3
From the last point, we know that L ( R n ) = H (Λ , (when G = 0 ) with uniformly equivalent norms. As in [16] , this can be improved:There exists a positive weight (cid:16) M ( α ; h ) ∈ S (1) such that if L ( dα ) = M ( α ; h ) L ( dα ) ( L being the Lebesgue measure), then ( u | v ) L ( R n ) = (cid:90) Λ T uT vL ( dα ) + ( Ku | v ) L ( R n ) , (5.17) where K is negligible of order 1 (as defined in the beginning of Subsection ) so that for every N ∈ N , K = O (1) : H (Λ , ( (cid:101) rR/h ) − N ) → H (Λ , ( (cid:101) rR/h ) N ) . Notice that the weight H is zero when G = 0 . .4 Pseudodifferential- and Fourier integral operators Such operators can be defined directly (cf. (6.3), (7.8) in [16]). We will onlyneed their descriptions on the FBI-side, somewhat in the spirit of Toeplitzoperators.Let m be an order function on Λ G . We say that R : H (Λ G , m ) → H (Λ G ,
1) is negligible of order m if for every order function (cid:101) m and every N ∈ N , R is a well defined operator H (Λ G , m (cid:101) m ) → H (Λ G , (cid:101) m ( (cid:101) rR/h ) N )which is uniformly bounded in the limit h →
0. (“Well defined” here refersto the existence of a unique extension from the dense subspace C ∞ ( R n ).)We have a completely analogous notion of negligible operators of order m : L (Λ G , m ) → L (Λ G , L (Λ G , m ) = L (Λ G , m e − H/h dα )for short. We will use the abbreviationnop = negligible operator,pop = pseudodifferential operator,top = Toeplitz operator.Let Π be the orthogonal projection L (Λ G , m ) → T H (Λ G , m ). Then (see[16], (7.24) and the adjacent discussion)Π = (cid:101) Π + Π −∞ , Π −∞ is L -negligible of order 1 , (cid:101) Π u ( α ) = (cid:90) p ( α, β ; h ) e ih ψ ( α,β ) u ( β ) m ( β ) e − H ( β ) /h dβ, (5.18)where ψ is independent of m and of class S ( (cid:101) rR ) in a region { ( α, β ); d g ( α, β ) ≤ / O (1) } and satisfies, − (cid:61) ψ ( α, β ) − H ( α ) − H ( β ) (cid:16) − (cid:18) (cid:101) rR | α x − β x | + R (cid:101) r | α ξ − β ξ | (cid:19) . (5.19)Moreover, p ∈ S ( m − h − n ) is supported in a region d g ( α, β ) ≤ / O (1) (5.20)and we have ψ ( α, β ) = − ψ ( β, α ) , p ( α, β ; h ) = p ( β, α ; h ) . (5.21)We refrain from recalling the characterization of T H (Λ G , m ) as the approxi-mate null space of a left ideal of pseudodifferential operators.We also have a class of pseudodifferential operators of order m ([16]) A : H (Λ G , (cid:101) m ) → H (Λ G , (cid:101) m/m ), ∀ (cid:101) m . Such an operator has an associated49rincipal symbol σ ( A ) ∈ S ( m, Λ G ) /S ( mh/ ( (cid:101) rR ) , Λ G ) which determines theoperator A up to an operator of order mh/ ( (cid:101) rR ) and the principal symbolmap is a bijection between the corresponding quotient spaces of operatorsand of symbols. We also have the usual result for the composition modulonegligible operators.When P is an h -differential operator as in (4.6)–(4.13) with coefficientsthat are holomorphic near π x supp G , then P is an h -pseudodifferential op-erator of order m = m ( x )( (cid:101) r ( x, ξ ) /r ( x )) N , associated to Λ G and the corre-sponding principal symbol is p | Λ G . (5.22)According to Proposition 7.3 in [16] the classes { Π b Π; b ∈ S ( m ) } and { T AT − Π; A is an h -pseudo of order m associated to Λ G } coincide modulonegligible operators of order m . Moreover, b and A are related by b ≡ σ A mod S (cid:18) mh (cid:101) rR (cid:19) . (5.23)Now, let G , G be two functions with the properties of G above. Then(see the beginning of Chapter 7 in [16]) there exists a smooth real bijectivecanonical transformation κ : Λ G → Λ G such that, writing ( x, ξ ) = κ ( y, η ),we have x − y ∈ S ( R ) , ξ − η ∈ S ( (cid:101) r ) , either as functions of ( y, η ) ∈ Λ G or of ( x, ξ ) ∈ Λ G . We can then de-fine Fourier integral operators of order m , associated to κ ; A = O (1): H (Λ G , (cid:101) m ) → H (Λ G , (cid:101) m/m ) , ∀ (cid:101) m . Such operators have the usual compo-sition result up to negligible operators. Moreover, we have the usual notionof elliptic operators: If U : H (Λ G , m ) → H (Λ G ,
1) is an elliptic Fourierintegral operator of order m , then (for h small enough) U is bijective andthe inverse is an elliptic Fourier integral operator of order m − associated to κ − up to a negligible operator of order m − . We also have a correspondingEgorov’s theorem: With U as above, let A be a pseudodifferential operatorof order (cid:98) m associated to Λ G . Then B = U − AU is a pseudodifferential op-erator of order (cid:98) m , associated to Λ G (up to a negligible operator of the sameorder), and the principal symbols are related by σ B = σ A ◦ κ. (5.24)We now specify the above in the case when G = 0 , G = G (5.25)50nd in doing so we go slightly beyond [16]. Since there will be several differentsymplectic frameworks, let us denote the standard real Hamilton field of G on R n , by H R n G . Recall thatΛ υG = { ρ ∈ C n ; (cid:61) ρ = υH R n G ( (cid:60) ρ ) } . (5.26)We let σ = (cid:80) n dξ j ∧ dx j denote the complex symplectic form on C nx × C nξ .The real and imaginarty parts (cid:60) σ and (cid:61) σ are real symplectic forms. When f is a real C function on some open subset of C n , we let H (cid:60) σf and H (cid:61) σf denotethe corresponding Hamilton fields. In general, if r = p + iq is differentiablewith complex-linear differential at some point, then at that point (cf. [33],(11.5), (11.6)), (cid:98) H r = H (cid:61) σq , J (cid:98) H r = H (cid:61) σp . (5.27)Here, J = multiplication of tangent vectors with i , H r denotes the complexHamilton field for σ (of type 1,0) and the hat indicates that we take thecorresponding real vector field; (cid:98) H r = H r + H r , H r = r (cid:48) ζ · ∂ z − r (cid:48) z · ∂ ζ .Returning to (5.26), if G ( ρ ) = G ( (cid:60) ρ ) is considered as a function on C n ,we have H (cid:61) σG = J H R n G . Then we can view the family Λ υG as obtained from Λ by deformation withthe field ν υ = H (cid:61) σG | Λ υG . Since Λ υG is I -Lagrangian and we get the same deformation is we modify ν υ by adding a field tangent to Λ υG , we can replace ν υ with (cid:101) ν υ = H (cid:61) σF υ | Λ υG , if F υ is real, smooth and F υ = G on Λ υG .Let (cid:101) G υ be an almost holomorphic extension from Λ υG of G | Λ υG . Then atΛ υG , J (cid:98) H (cid:101) G υ = H (cid:61) σ (cid:60) (cid:101) G υ ≡ H (cid:61) σG mod T Λ υG , by (5.27), so J (cid:98) H (cid:101) G υ generates the family Λ υG by deformation from Λ . (cid:101) G υ can be constructed in the following way: Consider the map θ = θ υ : R n (cid:51) ρ (cid:55)→ ρ + iυH R n G ( ρ ) =: ρ + iγ υ ( ρ ) ∈ C n . For k ∈ N , ∂ kυ γ υ is of class S (1) for the metric g . (Here we use that H G isof class S (1) for the metric g .) Thus ∂ kυ θ υ is of class ˙ S (1) when k = 0 and ofclass S (1) when k ≥ (cid:101) θ υ : C n → C n
51e an almost holomorphic extension of θ υ with the same symbol propertiesand let (cid:101) G ∈ S ( (cid:101) rR ) be an almost holomorphic extension of G . We notice that (cid:101) θ υ is a local diffeomorphism and that (cid:101) θ − υ : neigh (Λ υG , C n ) → C n is almostholomorphic at Λ υG with the same symbol properties. Then (cid:101) G υ := (cid:101) G ◦ (cid:101) θ − υ has the required properties. One can also see that it can be defined in a1 / O (1)-neighborhood of Λ υG for g , and be of class S ( (cid:101) rR ) there with all its t -derivatives. Using (cid:101) G υ , we get a smooth family of canonical transformations κ υ : Λ → Λ υG by integration of˙ κ υ ( ρ ) = H i (cid:101) G υ ( κ υ ( ρ )) , ρ ∈ Λ (identifying H i (cid:101) G υ (cid:39) J (cid:98) H (cid:101) G υ ) . In this way κ υ is defined in a 1 / O (1)-neighborhood of Λ for g and almostholomorphic at Λ . κ υ ∈ ˙ S (1), ∂ kυ κ υ ∈ S (1) for k ≥ G (cid:96)υ = (cid:101) G υ and let G rυ be the almost holomorphic function at Λ which is given by G (cid:96)υ ◦ κ υ = G rυ . (5.28)We have ∂ kυ G (cid:96)υ , ∂ kυ G rυ ∈ S ( (cid:101) rR ) for k ≥ . Then on Λ υG : H G (cid:96)υ = ( κ υ ) ∗ H G rυ , where ( κ υ ) ∗ denotes the operation of push forward of vector fields.Let G rυ , G (cid:96)υ be pseudodifferential operators of order (cid:101) rR associated to Λ ,Λ υG with principal symbols G rυ and G (cid:96)υ respectively. We can also assumethat ∂ kυ G rυ is a pseudodifferential operator of order (cid:101) rR for all k . Then wehave elliptic Fourier integral operators U υ , (cid:101) U υ of order 1 associated to κ υ ,such that hD υ U υ + iU υ G rυ = K rυ , U = 1 , (5.29) hD υ (cid:101) U υ + i G (cid:96)υ (cid:101) U υ = K (cid:96)υ , (cid:101) U = 1 , (5.30)where K rυ , ∂ kυ K rυ , K (cid:96)υ , ∂ kυ K (cid:96)υ are negligible operators of order (cid:101) rR . This is astraight forward WKB-solution of Cauchy problems within the framework of[16]. Now replace G rυ with G rυ + iU − υ K rυ and G (cid:96)υ with G (cid:96)υ + iK (cid:96)υ (cid:101) U − υ and noticethat U − υ K rυ and K (cid:96)υ (cid:101) U − υ are negligible of order 1 with all their υ -derivatives.Then we get, hD υ U υ + iU υ G rυ = 0 , U = 1 , (5.31) hD υ (cid:101) U υ + i G (cid:96)υ (cid:101) U υ = 0 , (cid:101) U = 1 . (5.32)If we choose first G rυ , U υ in (5.31) and then determine G (cid:96)υ by G (cid:96)υ U υ = U υ G rυ (5.33)52in formal agreement with Egorov’s theorem and (5.28)), we get (cid:101) U υ = U υ in(5.32): hD υ U υ + i G (cid:96)υ U υ = 0 , U = 1 . (5.34)Using also (5.42) below, we get( hD υ ) k G (cid:96)υ = U υ (cid:0) hD υ − i ad G rυ (cid:1) k ( G rυ ) U − υ , (5.35)which shows that for every k ≥ ∂ kυ G (cid:96)υ is the sum of a pseudodifferentialoperator and a negligible operator of order (cid:101) rR . Here ad A ( B ) denotes thecommutator [ A, B ].Let P be an h -differential operator of order m = m ( x )( (cid:101) r/r ) N as in(4.6)–(4.13), so that P is also an h -pseudodifferential operator P : H (Λ υG , m ) → H (Λ υG ,
1) (5.36)with principal symbol p | Λ υG as in (5.22). Here we also assume that thecoefficients of P are analytic in a neighborhood of the x -space projection ofsupp G . The study of P in (5.36) is equivalent to that of V υ P U υ =: P υ : H (Λ , m ) → H (Λ , , where V υ = U − υ . (5.37)We will often write H (Λ υG ) = H (Λ υG , P u | v ) H (Λ υG ) = ( P υ V υ u | V υ v ) , if we define the norm on H (Λ υG ) by (cid:107) v (cid:107) H (Λ υG ) = (cid:107) V υ v (cid:107) L , (5.38)making the operators U υ : L → H (Λ υG ) and V υ : H (Λ υG ) → L unitary.This norm is uniformly equivalent to the one in (5.16). Remark 5.4
Let Ω (cid:98) R n be open and assume that G ( x, ξ ) = 0 whenever x ∈ (cid:101) Ω , where (cid:101) Ω is a neighborhood of Ω . We can choose first the formalpseudodifferential operator part of G rυ with symbol equal to zero over (cid:101) Ω . Thenformally, U υ is a Fourier integral operator equal to 1 on L ( (cid:101) Ω) . It followsfrom the way Fourier integral operators are defined in [16] , that we can choosea realization of U υ (that we denote with the same symbol) such that U υ u = u, when u ∈ C ∞ (Ω) . (5.39) As before, let V υ = U − υ . Applying V υ to (5.39) , we get V υ u = u, when u ∈ C ∞ (Ω) . (5.40) After that we modify G (cid:96)υ , G rυ with negligible terms as above, so that (5.31) , (5.34) hold. From (5.38) , we now get (cid:107) v (cid:107) H (Λ υG ) = (cid:107) v (cid:107) L , v ∈ C ∞ (Ω) . (5.41)53rom (5.31), we first notice that hD υ V υ − i G rυ V υ = 0 , (5.42)and then that h∂ υ P υ = [ P υ , G rυ ] . (5.43)We already know that P υ = P (cid:48) υ + N υ where P (cid:48) υ , N υ are continuous in υ with values in the pseudodifferential and negligible operators respectively, oforder m . See the statements 1–3 after Theorem 7.2 in [16]. Write (5.43) as (cid:18) ∂ υ + 1 h ad G rυ (cid:19) P υ = 0 , which implies, (cid:18) ∂ υ + 1 h ad G rυ (cid:19) k P υ = 0 , k = 1 , , ... From this we deduce that ∂ kυ P υ has the same structure. From Taylor’s for-mula with integral remainder, we get P υ = P υ,k + N υ,k for every k ∈ N , where υ (cid:55)→ P υ,k , υ (cid:55)→ N υ,k are of class C k with values inthe pops of order m and nops of order m respectively.On the other hand, since the machineries are based on the (complex)method of stationary phase, we also know that the Weyl symbols of P υ and P υ,k are of the form ∼ ∞ (cid:88) h j p j ( υ, x, ξ ) , (5.44)where p j ∈ S ( m/ ( (cid:101) rR ) j ) are independent of k and therefore smooth in υ . Weconclude that P υ = P (cid:48) υ + N υ , where P (cid:48) υ , N υ are smooth in υ with values inthe pops and nops respectively, of order m .The equation for p ( υ, x, ξ ) = p υ ( x, ξ ) is ∂ υ p υ = iH G rυ p υ , p υ =0 = the principal symbol of P. We recover the fact (already known by Egorov’s theorem) that p υ ( ρ ) = p ( κ υ ( ρ )) =: (cid:101) p υ . (5.45)Indeed, the two symbols are equal when υ = 0 and ∂ υ (cid:101) p υ ( ρ ) = (cid:104) ˙ κ υ ( ρ ) , dp ( κ υ ( ρ )) (cid:105) = i (cid:104) ( κ υ ) ∗ H G rυ , dp ( κ υ ( ρ )) (cid:105) = (cid:104) H G rυ , κ ∗ υ ( dp ( κ υ ( ρ ))) (cid:105) = i (cid:104) H G rυ , d ( p ◦ κ υ ( ρ )) (cid:105) = iH G rυ (cid:101) p υ . κ υ prior to (5.28), we see that κ υ ( ρ ) = ρ + iυH G ( ρ ) + O ( υ ) , (5.46)where the remainder is O ( υ ) as a smooth function of υ with values in S (1)(with respect to the metric g ). Using this in (5.45), we get p υ ( ρ ) = p ( ρ ) − iυH p G + O ( υ m ) (5.47)in the sense of smooth functions neigh (0 , R ) (cid:51) υ (cid:55)→ S ( m ).From (5.46) we get (cid:101) ρ := (cid:60) κ υ ( ρ ) = ρ + O ( υ ) , so that by (5.26) κ υ ( ρ ) = (cid:101) ρ + iυH G ( (cid:101) ρ ) (5.48)and hence, p υ ( ρ ) = p ( (cid:101) ρ + iυH G ( (cid:101) ρ )) . (5.49)If G = G s ∈ S ( (cid:101) rR ) is real and depends smoothly on s ∈ neigh (0 , R ),then the smooth dependence on s diffuses into the whole construction aboveand we get (with the obvious notation) that P υ,s := V υ,s P U υ,s in (5.37) is asmooth function of ( υ, s ) with values in the pops+nops of order m . (Recallthat we sometimes abbreviate: pop=pseudodifferential operator, nop=negli-gible operator.)Let Π be the orthogonal projection L (Λ , M ) → T L ( R n ) (cf. Remark5.3) whose properties were recalled in (5.18)–(5.21). Combining the aboveproperties of P υ,s with Proposition 7.3 in [16], we get T P υ,s T − Π = Π P top υ,s Π + N υ,s (5.50)where N υ,s is smooth in ( υ, s ) with values in the nops of order m and P top υ,s ( ρ ; h ) ∼ ∞ (cid:88) h k p kυ,s ( ρ ) , ρ ∈ Λ , (5.51)in C ∞ (neigh (0 , , S ( m )) and with the general term in the sum belonging to C ∞ (neigh (0 , , h k S ( m/ ( (cid:101) rR ) k )). Here, as already recalled in (5.23), p υ,s = p υ,s (5.52)is the principal symbol of P υ,s .From (5.50) we infer that( P U υ,s u | U υ,s v ) H (Λ υGs ) = ( P υ,s u | v )= (cid:90) Λ P top υ,s ( ρ ; h ) T u ( ρ ) · T v ( ρ ) L ( dρ ) + ( N υ,s u | v ) , (5.53)55or u, v ∈ H (Λ , m / ) (cf. Remark 5.3 and (5.38)). This can be expressed inthe coordinates (cid:101) ρ in (5.48). Here the scalar product in the middle is the oneof L ( R n ). The Jacobian satisfies J υ,s ( (cid:101) ρ ) := dρd (cid:101) ρ = 1 + O ( υ ) in S (1) (5.54)and is a smooth function of υ, s . We can write P top υ,s ( ρ ; h ) = (cid:101) P top υ,s ( (cid:101) ρ ; h ) , (5.55)so (5.53) becomes( P υ,s u | v ) = (cid:90) Λ (cid:101) P top υ,s ( (cid:101) ρ ; h ) (cid:101) T u ( (cid:101) ρ ) · (cid:101) T v ( (cid:101) ρ ) J υ,s ( (cid:101) ρ ) L ( d (cid:101) ρ ) + ( N υ,s u | v ) , (5.56)where (cid:101) T u ( (cid:101) ρ ) := T u ( ρ ). (cid:101) P top υ,s ( (cid:101) ρ ; h ) has an asymptotic expansion as in (5.51)with (cid:101) p kυ,s ( (cid:101) ρ ) = p kυ,s ( ρ ) and the advantage with (5.56) is that (cid:101) p υ,s = (cid:101) p υ,s satisfies (cid:101) p υ,s ( (cid:101) ρ ) = p ( (cid:101) ρ + iυH G s ( (cid:101) ρ )) . (5.57)All this remains valid if we replace the single parameter s by s = ( s , ..., s k ) ∈ neigh(0 , R k ).If p is real-valued on Λ , we get (cid:61) (cid:101) p υ, ( (cid:101) ρ ) = υH G ( p ) + O ( mυ (cid:107) H G (cid:107) g )= − υH p ( G ) + O ( mυ (cid:107) H G (cid:107) g ) . (5.58)We summarize the results in this section. Proposition 5.5
Let P be an h -differential operator of order m ( x, ξ ) = m ( x )( (cid:101) r/r ) N as in (4.6) – (4.13) . Let G ∈ S ( (cid:101) rR ) satisfy (5.7) and assumethat the coefficients of P are analytic in a neighborhood of the x -space pro-jection of supp ( G − g ) . Then for ≤ υ ≤ , P : H (Λ υG , m ) → H (Λ υG , is the sum of an h -pop and a nop both of order m , depending smoothly on υ .The principal symbol is equal to p | Λ υG .We can find a canonical transformation κ υ : Λ → Λ υG of class ˙ S (1) forthe metric g , depending smoothly on υ ∈ [0 , in that class, satisfying (5.46) and an operator U υ : H (Λ , → H (Λ υG , of the form U (cid:48) υ + N υ , where U (cid:48) υ is an elliptic Fourier integral operator of order 1 associated to κ υ and N υ isa nop of order 1, with U = id , such that U − υ =: V υ = V (cid:48) υ + M υ has theanalogous properties (with κ υ replaced with κ − υ , such that P υ := V υ P U υ hasthe following properties: P υ is the sum of a pop and a nop of order m , both depending smoothlyon υ in the corresponding spaces of operators. • The principal symbol of P υ is given by (5.49) , (5.48) . • Writing the Weyl symbol of P υ as ∼ (cid:80) ∞ h j p j ( υ, x, ξ ) , we have supp ( p j ( υ, · ) − p j (0 , · )) ⊂ supp G, j ≥ . • We have the Toeplitz representation (5.50) , (5.51) , (5.53) (without theparameters s for the moment), where the leading symbol in (5.51) isequal to the one of P υ as a pseudodifferential operator, i.e. p ( υ, x, ξ ) .When G depends smoothly on additional parameters s ∈ neigh (0 , R k ) wehave the corresponding smooth dependence of all terms above.When G ∈ S ( m G ) satisfies the more special condition (5.8) , we can choose U υ so that P υ − P is of order mm G / ( (cid:101) rR ) . More precisely, ∂ υ P υ ∼ (cid:80) ∞ h j ∂ υ p j in S ( mm G / ( (cid:101) rR )) , ∂ υ p j ∈ S ( mm G / ( (cid:101) rR ) j +1 ) . A similar statement holds for P top υ,s , (cid:101) P top υ,s and we here retain that ∂ t ( P top υ,s − p υ,s ) ∈ S ( hmm G / ( (cid:101) rR ) ) . The extension in the last paragraph of the proposition follows from aninspection of the proofs. H (Λ υG ) spaces We continue the discussion from the preceding section and work with therepresentations (5.56), (5.57), where we drop the tildes until further notice.From (5.57), we get ∂ s p υ,s ( ρ ) = iυ (cid:104) H ∂ s G s ( ρ ) , dp ( ρ + iυH G s ( ρ )) (cid:105) , which can also be written ∂ s p υ,s ( ρ ) = − iυ (cid:104) H p ( ρ + iυH G s ( ρ )) , d∂ s G s ( ρ ) (cid:105) , (6.1)If p ∈ S ( m ) is real-valued on Λ , we get by Taylor expansion “in the g -metric”, ∂ s (cid:61) p υ,s ( ρ ) = − υ (cid:104) H p ( ρ ) , d∂ s G s ( ρ ) (cid:105) + O (cid:16) υ m (cid:101) rR (cid:107) H G s (cid:107) g (cid:107) d∂ s G s (cid:107) g ∗ (cid:17) . (6.2)Here the factor m/ ( (cid:101) rR ) corresponds to the estimate (4.28) and g ∗ is the dualmetric to g : g ∗ = ( (cid:101) rdξ ) + ( Rdx ) , (6.3)57o (cid:107) df (cid:107) g ∗ = (cid:101) r | ∂ ξ f | + R | ∂ x f | = O ( (cid:101) r R ) , when f ∈ ˙ S ( (cid:101) rR ). Hence, if we assume a uniform bound on ∂ s G s in S ( (cid:101) rR ),(6.2) simplifies to ∂ s (cid:61) p υ,s ( ρ ) = − υH p ( ∂ s G s )( ρ ) + O ( mυ (cid:107) H G s (cid:107) g ) . (6.4)In this formula we can take s = ( s , ..., s k ) close to 0 in R k and replace ∂ s with ∂ s k : ∂ s k (cid:61) p υ,s ( ρ ) = − υH p ( ∂ s k G s )( ρ ) + O ( mυ (cid:107) H G s (cid:107) g ) . Taylor expansion at s = 0 gives, ∂ s k (cid:61) p υ,s ( ρ ) = − υH p ( ∂ s k G s ) s k =0 ( ρ )+ O ( mυs k )+ O ( mυ | s | )+ O ( mυ (cid:107) H G (cid:107) g ) . (6.5)Writing s = ( s (cid:48) , s k ) and integrating (6.5) from 0 to s k , gives − (cid:61) p υ,s ( ρ ) + (cid:61) p υ, ( s (cid:48) , ( ρ ) = υs k H p ( ∂ s k ) s k =0 G s + O ( mυs k ) + O ( mυ | s | s k ) + O ( mυ s k (cid:107) H G ( ρ ) (cid:107) g ) . (6.6)We now consider the situation in Proposition 4.2. Let K ⊃ K ⊃ K ⊃ ... be a sequence of compact H p -convex sets in Σ [ − (cid:15) ,(cid:15) ] that contain the trappedset. We choose K j so that K j +1 is contained in the interior of K j . For j ∈ N , let χ j ∈ C ∞ (] − (cid:15) , (cid:15) [; [0 , − (cid:15) / , (cid:15) /
2] and suchthat χ j +1 = 1 on supp χ j .Let G be a modification of G on a bounded subset of Σ [ − (cid:15) ,(cid:15) ] as “ (cid:101) G ” inProposition 4.2 with (cid:101) K there equal to K . Let G j be constructed similarlywith (cid:101) K equal to K j in such a way that H p G j +1 > [ − (cid:15) ,(cid:15) ] \ K j . Thisimplies that H p G j +1 > G j .Let G j = χ j ( p ) G j . Since H p G j = χ j ( p ) H p G j , we get H p G j +1 ≥ m O (1) on supp G j . (6.7)Let G ( N ) = G + hG + ... + h N G N and notice that this enters into the framework of Section 5: G ( N ) = G s = G + s G + ... + s N G N , s = ( h, h , ..., h N ) .
58e apply (5.56) (with the tildes dropped since the beginning of this section)with υ > G ( k − to G ( k ) , the leading term p υ,s changesaccording to (6.6). Writing p ( k ) υ = p υ, ( h,..,h k , ,.., , we get − (cid:61) p ( k ) υ + (cid:61) p ( k − υ = υh k H p ( G k ) + O ( mυh k ) + O ( mυ h k +1 ) + O ( mυ h k (cid:107) H G (cid:107) g )= υh k H p ( G k ) + O ( mυh k +1 ) + O ( mυ h k (cid:107) H G (cid:107) g ) . (6.8)Also, by (5.58), −(cid:61) p υ, = υH p ( G ) + O ( mυ (cid:107) H G (cid:107) g )= υ (1 + O ( υ )) H p ( G ) , (6.9)where we also used (4.58) for G = G . Using that estimate also for the lastterm in (6.8) and summing over k , we get −(cid:61) p ( N ) υ = υ (cid:0) O (0) ( υ ) + O (1) ( υ h ) + ... + O ( N ) ( υ h N ) (cid:1) H p ( G )+ υ (cid:0) hH p G + h H p G + ... + h N H p G N (cid:1) + (cid:0) mυ O (1) ( h ) + mυ O (2) ( h ) + ... + mυ O ( N ) ( h N +1 ) (cid:1) . (6.10)Here O ( k ) ( · ) denotes a term which depends on G , ..., G k but not on G k +1 , ... and whose support is contained in that of G k . We see that after successivereplacements, G j (cid:55)→ α j G j with α j > h H p G + mυ O (1) ( h ) ≥ ,h H p G + mυ O (2) ( h ) ≥ ,.... − (cid:61) p ( N ) υ ≥ υ (1 + O ( υ )) H p ( G ) − O ( υmh N +1 ) . (6.11)Now recall (5.51) (after adding and removing the tildes), where p kυ,s arereal for υ = 0. Taylor expand each p kυ,s to sufficiently high order at s = 0and take s = ( h, h , ..., h N ). Then we get with P top , ( N ) υ = P top υ, ( h,..,h N ) , − (cid:61) P top , ( N ) υ ( ρ ; h ) = − (cid:61) p ( N ) υ + h O (0) ( mυ ) + h O (1) ( mυ ) + ... + h N +1 O ( N ) ( mυ ) . (6.12)Here the factors O ( j ) belong to S ( mυ ). They are independent of h for j ≤ N −
1. By successive replacements G j (cid:55)→ α j G j , we can achieve, using (6.10),that − (cid:61) P top , ( N ) υ ≥ υ (1 + O ( υ )) H p ( G ) − O ( υmh N +1 ) . (6.13)59ence, by (5.56), −(cid:61) ( P ( N ) υ u | u ) ≥ (cid:90) Λ υ (1 + O ( υ )) H p ( G ) (cid:107) T u ( ρ ) (cid:107) J υ, ( h,..,h N ) L ( dρ ) − O ( υ ) h N +1 (cid:107) u (cid:107) H (Λ ,m / ) . (6.14)The replacement G j (cid:55)→ α j G j does not depend on the value of N ≥ j , sowe get a full sequence G , G , ... . Consider an asymptotic sum G ∼ ∞ (cid:88) G j h j in S ( (cid:101) rR ) . (6.15)Then for every N ≥ G = G ( N ) + h N +1 (cid:101) G N +1 , (cid:101) G N +1 ∈ S ( (cid:101) rR ) and if P υ = V υ P U υ (with the natural definitions of U υ and V υ = U − υ ) we have theanalogue of (5.56) (now with the tildes dropped),( P υ u | v ) = (cid:90) Λ P top υ ( ρ ; h ) T u ( ρ ; h ) · T v ( ρ ; h ) J υ ( ρ ; h ) L ( dρ ) + ( N υ u | v ) . (6.16)Here N υ is negligible of order m , (cid:61) P top υ and (cid:61) N υ vanish for υ = 0. We canreplace P ( N ) υ with P υ in (6.14) and the discussion leading to that estimateshows that0 < J υ ( ρ ; h ) = 1+ O ( υ ) , −(cid:61) P top υ ≥ υ (1+ O ( υ )) H p G −O ( mυh ∞ ) . (6.17)In particular, − (cid:61) ( P υ u | u ) ≥ − υ O ( h ∞ ) (cid:107) u (cid:107) H (Λ ,m / ) . (6.18)Recall that P υ is just a reduction to H (Λ ) of the restriction to H (Λ υG , m )of P , so with the norm and scalar product on H (Λ υG ) induced by U υ , weget − (cid:61) ( P u | u ) H (Λ υG , ≥ − υ O ( h ∞ ) (cid:107) u (cid:107) H (Λ υG ,m / ) , (6.19)for u ∈ H (Λ υG , m ). Remark 6.1
Only m | Σ [ − (cid:15) ,(cid:15) matters in the calculations. Especially, in theSchr¨odinger case (discussed in Section ), we have m = (cid:104) ξ (cid:105) , so we canreplace m with . We end this section with an observation about decoupling of the exteriorand the interior part in certain situations. Let G be as in (6.15). Let Ω ⊂ R n be a bounded open set such thatΩ ∩ π x (supp G ) = ∅ , π x ( x, ξ ) = x. (6.20)60e have seen in Remark 5.4 that we can choose the Fourier integral operators U υ , V υ in Section 5 so that (5.39)–(5.41) hold for G = G : (cid:107) v (cid:107) H (Λ υG ) = (cid:107) v (cid:107) L , v ∈ C ∞ (Ω) . (6.21)It follows that −(cid:61) ( P u | u ) H (Λ υG ) = −(cid:61) ( (cid:101) P u | u ) H (Λ υG ) if (cid:101) P is a new formally self-adjoint operator (with respect to L ( R n )) suchthat supp ( (cid:101) P − P ) ⊂ Ω, where supp ( (cid:101) P − P ) is defined to be the union of thesupports of the coefficients of (cid:101) P − P . In particular, we may then replace P with (cid:101) P in (6.19). In this section we discuss improvements in the semi-bound estimates, whenthe escape function is supported far away. We let P , m , r , (cid:101) r , R be as inSection 4 with the following special choices, r = 1 , R ( x ) = (cid:104) x (cid:105) , m ( x ) = 1 , (7.1)implying m ( x, ξ ) = (cid:104) ξ (cid:105) N , (cid:101) r ( x, ξ ) = (cid:104) ξ (cid:105) . (7.2)With p ( x, ξ ) still denoting the semi-classical principal symbol, we assumethat p ( x, ξ ) → p ∞ ( ξ ) ∈ S ( m ) as x → ∞ in the following sense: For all α, β ∈ N n , ∂ αx ∂ βξ ( p ( x, ξ ) − p ∞ ( ξ )) = o (1) m ( ξ ) R ( x ) −| α | (cid:101) r ( x, ξ ) −| β | , x → ∞ , (7.3)uniformly with respect to ξ .If we assume the existence of an escape function in Σ [ − (cid:15) ,(cid:15) ] for (cid:15) > p ∞ ( ξ ) = 0 ⇒ ∂ ξ p ∞ ( ξ ) (cid:54) = 0 . (7.4)From the ellipticity assumption (4.14) we know in addition to (4.15), (4.16)that p − ∞ (0) is bounded. Conversely, if we assume (7.4), then G ( x, ξ ) = x · ∂ ξ p ∞ ( ξ ) (cid:104) ξ (cid:105) N − ∈ S ( R (cid:101) r )has the required properties. Indeed, when x → ∞ , H p G ( x, ξ ) → H p ∞ G = ( ∂ ξ p ∞ ) (cid:104) ξ (cid:105) N − (cid:16) m on p − ∞ (0) .
61n the classical Schr¨odinger operator case, this gives G ( x, ξ ) = 2 x · ξ , which upto the factor 2 is the escape function appearing in standard complex scaling.We study the situation in a domain | x | > µ/ O (1), for µ (cid:29)
1, and even-tually we will choose our escape function G with its support contained insuch domains. It is natural to make the change of variables, x = µ (cid:101) x , so that | (cid:101) x | > / O (1).Consider first the principal symbol. Put p µ ( (cid:101) x, (cid:101) ξ ) = p ( µ (cid:101) x, (cid:101) ξ ) = p ◦ κ µ ( (cid:101) x, (cid:101) ξ ) , (7.5)where κ µ ( (cid:101) x, (cid:101) ξ ) = ( µ (cid:101) x, (cid:101) ξ ) , (7.6) κ ∗ µ σ = µσ. (7.7)Then in any region, | (cid:101) x | > / O (1) we have ∂ α (cid:101) x ∂ β (cid:101) ξ p µ ( (cid:101) x, (cid:101) ξ ) = O (1) m ( (cid:101) ξ ) (cid:101) r −| β | (cid:98) R ( (cid:101) x ) −| α | , (7.8) ∂ α (cid:101) x ∂ β (cid:101) ξ ( p µ ( (cid:101) x, (cid:101) ξ ) − p ∞ ( (cid:101) ξ )) = o (1) m ( (cid:101) ξ ) (cid:101) r −| β | (cid:98) R ( (cid:101) x ) −| α | , µ → ∞ (7.9)Here (cid:98) R = (cid:98) R µ is given by (cid:98) R ( (cid:101) x ) = R ( µ (cid:101) x ) µ , (7.10)so that (cid:98) R ( (cid:101) x ) (cid:16) R ( (cid:101) x ) , | (cid:101) x | > / O (1) . We restrict the attention to a region,Σ µ,(cid:15) = p − µ ([ − (cid:15) , (cid:15) ]) . (7.11) Proposition 7.1
The “balls” π − (cid:101) x B (0 , r ) ∩ Σ µ,(cid:15) are H p µ -convex for r ≥ / O (1) when µ is large enough. More precisely, every H p µ -trajectory in Σ µ,(cid:15) can visit such a ball only during at most one time interval which can be finiteor infinite. Proof.
It suffices to check that H p µ ( (cid:101) x / > , ( (cid:101) x, (cid:101) ξ ) ∈ Σ µ,(cid:15) , | (cid:101) x | ≥ / O (1) . H p µ (cid:18) (cid:101) x (cid:19) = (cid:18) ∂p µ ∂ (cid:101) ξ · ∂∂ (cid:101) x − ∂p µ ∂ (cid:101) x · ∂∂ (cid:101) ξ (cid:19) (cid:18) ∂p µ ∂ (cid:101) ξ · (cid:101) x (cid:19) = (cid:18) ∂p µ ∂ (cid:101) ξ (cid:19) + ∂p µ ∂ (cid:101) ξ · ∂ p µ ∂ (cid:101) x∂ (cid:101) ξ · (cid:101) x − ∂p µ ∂ (cid:101) x · ∂ p µ ∂ (cid:101) ξ · (cid:101) x → (cid:18) ∂p ∞ ∂ (cid:101) ξ (cid:19) > , µ → ∞ . (cid:50) From this proposition and (7.20) below, it will follow that the “balls” π − x ( B (0 , µ )) ∩ Σ (cid:15) are H p convex for µ large enough.We next apply the change of variables x = µ (cid:101) x to the operator P in (4.6).We get, P ( x, hD x ; h ) = P ( µ (cid:101) x, (cid:101) hD (cid:101) x ; h ) =: P µ ( (cid:101) x, (cid:101) hD (cid:101) x ; (cid:101) h ) , (cid:101) h = hµ . (7.12)More explicitly, in view of (4.6), (4.7): P µ ( (cid:101) x, (cid:101) hD (cid:101) x ; (cid:101) h ) = (cid:88) | α |≤ N a µα ( (cid:101) x ; (cid:101) h )( (cid:101) hD (cid:101) x ) α , (7.13)where, a µα ( (cid:101) x ; (cid:101) h ) = a α ( µ (cid:101) x ; h ) = N −| α | (cid:88) k =0 h k a α,k ( µ (cid:101) x ) = N −| α | (cid:88) k =0 (cid:101) h k a µα,k ( (cid:101) x ) . (7.14)Here a µα,k = µ k a α,k ( µ (cid:101) x ) ∈ S ( (cid:98) R − k ) , | (cid:101) x | ≥ / O (1) , (7.15)and (cid:98) R (cid:16) R ( (cid:101) x ) as in (7.10). This means that P µ satisfies the general as-sumptions for P in the region, | (cid:101) x | ≥ / O (1) and we have the analogue of(4.9): P µ ( (cid:101) x, (cid:101) ξ ; (cid:101) h ) = p ,µ ( (cid:101) x, (cid:101) ξ ) + (cid:101) hp ,µ ( (cid:101) x, (cid:101) ξ ) + ... + h N p N ,µ ( (cid:101) x, (cid:101) ξ ) , (7.16) p ,µ ( (cid:101) x, (cid:101) ξ ) = p µ ( (cid:101) x, (cid:101) ξ ), p j,µ ( (cid:101) x, (cid:101) ξ ) ∈ S ( m ( (cid:101) rR ) − j ) , | (cid:101) x | ≥ / O (1) , (7.17)63e next check that Λ υG scales naturally when G is an escape function.We expect the scaled weight G µ to satisfy, e υG ( x,ξ ) /h = e υG µ ( (cid:101) x, (cid:101) ξ ) / (cid:101) h , ( x, ξ ) = κ µ ( (cid:101) x, (cid:101) ξ ) , i.e. G µ ( (cid:101) x, (cid:101) ξ ) = G ( x, ξ ) /µ , so we define: G µ ( (cid:101) x, (cid:101) ξ ) = 1 µ G ( µ (cid:101) x, (cid:101) ξ ) = 1 µ ( G ◦ κ µ )( (cid:101) x, (cid:101) ξ ) . (7.18)We have, Λ υG µ = κ − µ (Λ υG ) . (7.19)Indeed, for ( (cid:101) x, (cid:101) ξ ) ∈ Λ υG µ , we have µ (cid:61) (cid:101) x = µυ∂ (cid:101) ξ G µ ( (cid:60) (cid:101) x, (cid:60) (cid:101) ξ ) = υ∂ ξ G ( µ (cid:60) (cid:101) x, (cid:60) (cid:101) ξ ) , (cid:61) (cid:101) ξ = − υ∂ (cid:101) x G µ ( (cid:60) (cid:101) x, (cid:60) (cid:101) ξ ) = − υ∂ x G ( µ (cid:60) (cid:101) x, (cid:60) (cid:101) ξ ) , which shows that ( x, ξ ) ∈ Λ υG if ( x, ξ ) = κ µ ( (cid:101) x, (cid:101) ξ ).In the same spirit, we observe that( κ µ ) ∗ H p µ = µH p . (7.20)We finally apply the natural scalingΛ υG (cid:51) α (cid:55)→ (cid:101) α = κ − µ ( α ) ∈ Λ υG µ to T u in (5.5), (5.6). Starting from (5.6), we put u ( y ) = (cid:101) u ( (cid:101) y ), where y = µ (cid:101) y .Again, with (cid:101) h = h/µ , we get from (5.3):1 h φ ( α, y ) = 1 (cid:101) h (cid:101) φ ( (cid:101) α, (cid:101) y ) , where (cid:101) φ ( (cid:101) α, (cid:101) y ) = ( (cid:101) α x − (cid:101) y ) · (cid:101) α ξ + iλ µ ( (cid:101) α )( (cid:101) α x − (cid:101) y ) (7.21)and λ µ ( (cid:101) α ) = µλ ( µ (cid:101) α x , (cid:101) α ξ ) ∈ S ( (cid:101) r ( (cid:101) α ) R ( (cid:101) α x ) − ) (7.22)for | (cid:101) α x | ≥ / O (1), where we also used that µ/R ( µ (cid:101) α x ) (cid:16) /R ( (cid:101) α x ). With thesame changes of variables in (5.6), we get T u ( α ; h ) = (cid:90) e i (cid:101) h (cid:101) φ ( (cid:101) α, (cid:101) y ) µ n t ( µ (cid:101) α x , (cid:101) α ξ , µ (cid:101) y ; µ (cid:101) h ) χ (cid:32) (cid:101) y − (cid:60) (cid:101) α x (cid:98) R ( (cid:101) α x ) (cid:33) (cid:101) u ( (cid:101) y ) d (cid:101) y, (cid:98) R (cid:16) R as in (7.10), so the cutoff is the naturally scaled one. Thenew amplitude (cid:101) t ( (cid:101) α, (cid:101) y ; (cid:101) h ) = µ n t ( µ (cid:101) α x , (cid:101) α ξ , µ (cid:101) y ; µ (cid:101) h ) , belongs to S (cid:16) µ n h − n (cid:101) r ( (cid:101) α ξ ) n R ( µ (cid:101) α x ) − n (cid:17) = S (cid:16)(cid:101) h − n (cid:101) r ( (cid:101) α ξ ) n (cid:98) R ( (cid:101) α x ) − n (cid:17) , which is the right symbol class (working still in | (cid:101) α x | ≥ / O (1)).Furthermore, (cid:12)(cid:12) det (cid:0)(cid:101) t ∂ (cid:101) y (cid:101) t ... ∂ (cid:101) y n (cid:101) t (cid:1)(cid:12)(cid:12) = µ ( n +1) n + n (cid:12)(cid:12) det (cid:0) t ∂ y t ... ∂ y n t (cid:1)(cid:12)(cid:12) (cid:16) (cid:98) R − n (cid:16)(cid:101) h − n (cid:101) r n (cid:98) R − n (cid:17) n +1 , which is analogous to (5.5).In conclusion, for | α x | ≥ µ/ O (1), we have T u ( α ; h ) = (cid:101) T (cid:101) u ( (cid:101) α ; (cid:101) h ) , Λ υG (cid:51) α = κ µ ( (cid:101) α ) , (cid:101) α ∈ Λ υG µ , (cid:101) u ( (cid:101) y ) = u ( y ) , y = µ (cid:101) y, (cid:101) h = h/µ, (7.23)where (cid:101) T has all the general properties of an FBI-transform in any fixed region | (cid:101) α x | ≥ / O (1).If the two order functions m and (cid:101) m are related by (cid:101) m = m ◦ κ µ , (7.24)which is fulfilled under the assumptions (7.1), (7.2), when (cid:101) m = (cid:104) (cid:101) ξ (cid:105) N , thenwe can define the Sobolev spaces H (Λ υG , m ), H (Λ υG µ , (cid:101) m ) as in Section 5.3,by (5.16), with G replaced by υG , and its analogue, (cid:107) (cid:101) u (cid:107) H (Λ υGµ , (cid:101) m ) = (cid:107) (cid:101) T (cid:101) u (cid:107) L (Λ υGµ , (cid:101) m e − υHµ/ (cid:101) h d (cid:101) α ) . (7.25)Here we use H in (5.16) adapted to G (cf. (5.2)) so that υH is adapted to υG ,define H µ by the analogous relation and notice that (cid:101) H µ ( (cid:101) x, (cid:101) ξ ) = µ − H ( µ (cid:101) x, (cid:101) ξ ).( G and H also depend on µ and we use the subscript µ to indicate when wework in the scaled variables ( (cid:101) x, (cid:101) ξ ).) If we extend the definition of κ µ to maps: R n → R n by putting κ µ ( (cid:101) y ) = µ (cid:101) y , and let κ ∗ µ denote right composition with κ µ in the usual way, then (7.23) tells us that κ ∗ µ ◦ T = (cid:101) T ◦ κ ∗ µ . dα = µ n d (cid:101) α , dy = µ n d (cid:101) y , so (cid:107) u (cid:107) L = µ n (cid:107) (cid:101) u (cid:107) L , (cid:107) T u (cid:107) L (Λ υG ,m e − υH/h dα ) = µ n (cid:107) (cid:101) T (cid:101) u (cid:107) L (Λ υGµ , (cid:101) m e − υHµ/h d (cid:101) α ) . (7.26)Thus, (cid:107) u (cid:107) H (Λ υG ,m ) = µ n (cid:107) (cid:101) u (cid:107) H (Λ υGµ , (cid:101) m ) . (7.27)This applies in particular to the spaces H (Λ υG ), H (Λ υG µ ).Now we apply the discussion at the end of Section 6 to the operator P µ ,whose symbol properties we have verified in any region | (cid:101) x | ≥ / O (1). In viewof Proposition 7.1 we have the strictly decreasing sequence of H p µ -convex setsin Σ µ,(cid:15) : (cid:101) K j = π − (cid:101) x ( B (0 , r j )) ∩ Σ µ,(cid:15) , j = 0 , , , ... (7.28)where 3 / > r > r > r , ... (cid:38) j → ∞ . This gives rise to a weight G µ ,vanishing over a neighborhood of B (0 , / − (cid:61) ( P µ (cid:101) u | (cid:101) u ) H (Λ υG µ ) ≥ − υ O ( (cid:101) h ∞ ) (cid:107) (cid:101) u (cid:107) H (Λ υG µ , (cid:101) m / ) , (7.29)had it been true that P µ is a differential operator of the right symbol class alsoinside a region | (cid:101) x | ≤ / O (1). However, this symbol property is guaranteedonly outside such balls, but according to the observation at the end of Section6, we can choose the H (Λ υG µ ) norm, so that −(cid:61) ( P µ u | u ) H (Λ υG µ ) = −(cid:61) ( (cid:101) P µ u | u ) H (Λ υG µ ) , whenever supp ( (cid:101) P µ − P µ ) is contained in some small fixed neighborhoodof B (0 , / (cid:101) P µ satisfying (7.29). Hence(7.29) holds for P µ . (It suffices to take (cid:101) P µ = (1 − χ ) P µ (1 − χ ), where χ ∈ C ∞ ( B (0 , / R ) is equal to 1 on B (0 , / P , we put G = µG µ ◦ κ − µ : G ( x, ξ ) = µG µ ( x/µ, ξ ) . Then (7.29) gives, − (cid:61) ( P u | u ) H (Λ υG ) ≥ − υ O (( h/µ ) ∞ ) (cid:107) u (cid:107) H (Λ υG ,m / ) . (7.30)By Remark 6.1, we can replace m by 1 in the Schr¨odinger case. In (7.29),the semi-classical parameter is (cid:101) h = h/µ while in (7.30) we are back to using h . 66 Resolvent estimates
To fix the ideas, we assume right away that n = 1 and that P = − h ∂ x + V ( x ) , V ∈ C ∞ ( R ; R ) . (8.1)We adopt the general assumptions of Section 7. More precisely, we assume(7.1): r = 1, R ( x ) = (cid:104) x (cid:105) , m ( x ) = 1 and (7.2) with N = 2 so that m ( x, ξ ) = (cid:104) ξ (cid:105) . Recall that (cid:101) r ( x, ξ ) = (cid:104) ξ (cid:105) . We also assume (7.3) with p ( x, ξ ) = ξ + V ( x ), p ∞ ( ξ ) = ξ , which amounts to ∂ αx V ( x ) = o (1) (cid:104) x (cid:105) −| α | , x → ∞ . (8.2)We also assume dilation analyticity near ∞ : V has a holomorphic extension to { x ∈ C ; (cid:60) x > C, |(cid:61) x | < |(cid:60) x | /C } and denoting the extension also by V, we have V ( x ) = o (1) , when x → ∞ in the truncated sector above. (8.3)The earlier discussion was focused on the energy level E = 0. Here we willapply it with P replaced by P − E for E ∈ [ E − , E + ] for 0 < E − < E + < + ∞ .In other terms we will mainly work in p − ([ E − , E + ]) , < E − < E + < + ∞ , (8.4)and the slight difference with the earlier discussion is that we now take awider energy range [ E − , E + ] instead of [ − (cid:15) , (cid:15) ].Assume that for a choice E ∈ ] E − , E + [,the H p -flow is non trapping in every unboundedconnected component of p − ( E ) . (8.5)Later, we shall strengthen this assumption to non-trapping in p − ( E ). Usingthe special structure of the symbol p ( x, ξ ) = ξ + V ( x ), we see that everyunbounded connected component Σ (cid:48) E of p − ( E ) is a simple smooth integralcurve γ : R (cid:51) t (cid:55)→ γ ( t ) = ( x ( t ) , ξ ( t )) of H p with one of the following 4properties:1) tξ ( t ) > x ( t ) → + ∞ , when | t | → ∞ ,2) tξ ( t ) < x ( t ) → −∞ , when | t | → ∞ ,3) ξ ( t ) > x ( t ) → ±∞ , when t → ±∞ ,67) ξ ( t ) < x ( t ) → ∓∞ , when t → ±∞ .Moreover, the union Σ E of all unbounded components of p − ( E ) is the unionof two different components as above where eitherI) One is of type 1) and the other is of type 2),orII) one is of type 3) and the other is of type 4)Using a sequence of cutoffs, χ j ( p ), χ j ∈ C ∞ (] E − , E + [), where1 [ E − + δ,E + − δ ] ≺ χ ≺ χ ≺ ..., a corresponding sequence of escape functions G , G , G , ... with G ≺ G ≺ G ≺ ... and a dilation (cid:101) x (cid:55)→ µ (cid:101) x , µ ≥
1, we obtain as in Sections 6, 7 afunction G = G µ of class S ( (cid:101) rR ), uniformly with respect to µ , with supportin Σ [ E − ,E + ] ∩ { ( x, ξ ); | x | ≥ µ/ (2 C ) } such that we have the semi-boundednessproperty (7.30) for 0 ≤ υ (cid:28) µ ≥ H p G ≥ χ ( p ) /C, on {| x | ≥ µ/C } . (8.6)(The only difference with Sections 6, 7 is that we have replaced [ − (cid:15) , (cid:15) ]with [ E − , E + ] which is quite straight forward in the Schr¨odinger case.) Sincewe shall next turn to resolvent estimates with more escape functions, it isconvenient to rename G : G µ sbd = G sbd := G . (8.7)For the resolvent estimates, we need to supply a suitable escape functionin the set Σ E ∩ {| x | ≤ µ/C } and to merge it to G µ sbd . Here we assume that E ∈ [ E − + δ, E + − δ ].First we can find an escape function G ∈ C ∞ (Σ E ; R ) (8.8)of class S ( (cid:101) rR ) such that H p G > , (8.9) G ( x, − ξ ) = − G ( x, ξ ) , (8.10) G ( x, ξ ) = x · ξ, | x | (cid:29) . (8.11)Observe that, since we are in the 1D case, (cid:104) G ( x, ξ ) (cid:105) (cid:16) (cid:104) x (cid:105) , (8.12)68o (cid:104) G (cid:105) and (cid:104) x (cid:105) are equivalent weights.Let 0 ≤ ϑ (cid:28)
1, to be fixed small enough. Let f = f ϑ ∈ C ∞ ( R ; R ) begiven by f (0) = 0 , f (cid:48) ( t ) = h (cid:104) t (cid:105) ϑ . (8.13)Then f is odd, and when ϑ >
0, we have f ( t ) = h ( ± C ϑ + O ( (cid:104) t (cid:105) − ϑ ) , t → ±∞ . (8.14)Here, C ϑ = (cid:90) + ∞ (cid:104) t (cid:105) ϑ dt. When ϑ = 0, we get f ( t ) = h (ln t + O (1)) , t → + ∞ . (8.15)Because of the unboundedness in this case, we assume from now on that ϑ > G by G = f ( G ) . (8.16)Then, H p G = f (cid:48) ( G ) H p G (cid:16) h (cid:104) G (cid:105) ϑ (cid:16) h (cid:104) x (cid:105) ϑ . (8.17)Also, H G = f (cid:48) ( G ) H G = h (cid:104) G (cid:105) ϑ H G and recalling that (cid:107) H G (cid:107) g = O (1), we get (cid:107) H G (cid:107) g = O ( h ) (cid:104) x (cid:105) ϑ . (8.18)This also follows from (8.19) below. Proposition 8.1
We have G ∈ ˙ S (cid:18) h (cid:104) x (cid:105) ϑ (cid:19) , (8.19) G ∈ S ( h ) . (8.20) Proof.
For k ≥
1, we have f ( k ) ( t ) = O ( h ) (cid:104) t (cid:105) − ϑ − k α, β ) ∈ N \
0, we can write ∂ αx ∂ βξ G as a finite linear combinationof terms f ( k ) ( G ) (cid:16) ∂ α x ∂ β ξ G (cid:17) ... (cid:16) ∂ α k x ∂ β k ξ G (cid:17) , (8.21)where k ≥
1, ( α j , β j ) (cid:54) = (0 , α + ... + α k = α , β + ... + β k = β . Since (cid:104) G (cid:105) (cid:16) (cid:104) x (cid:105) , it follows that the term in (8.21) is O (1) h (cid:104) x (cid:105) ϑ + k (cid:104) x (cid:105) − α ... (cid:104) x (cid:105) − α k = O ( h ) (cid:104) x (cid:105) ϑ + α and (8.19) follows. Now (8.20) follows from (8.19) and the fact that G = O ( h ) by (8.14). (cid:50) Until further notice, we assume thatThe H p flow is non-trapping on p − ( E ) . (8.22)In other words, Σ E is equal to all of p − ( E ). Let χ ∈ C ∞ (neigh ( E, R ); [0 , E and with its support contained in χ − (1), where χ , χ , ... are the cutoffs used before. Put G lap = χ ( p ) G ∈ S ( h ) . (8.23)Here “lap” stands for “limiting absorption principle”, because G lap can beused to give a quick proof of the semi-classical limiting absorption principleof Robert–Tamura [31], also proved by G´erard–Martinez [12]. This is alsorelated to Martinez’ result [23] on the absence of resonances for non-trappingpotentials that are merely smooth on some bounded set. We put G (cid:15) = G lap + (cid:15)G sbd , < (cid:15) (cid:28) h, (8.24)where we recall that G sbd depends on a large parameter µ . We next choose µ as a function of (cid:15) , and to do so we notice that when the support of χ isnarrow enough, H p G lap (cid:16) χ ( p ) h (cid:104) G (cid:105) ϑ (cid:16) χ ( p ) h (cid:104) x (cid:105) ϑ , (8.25)which is (cid:16) h/ (cid:104) x (cid:105) ϑ where χ ( p ) = 1 and in particular in p − ([ E − δ , E + δ ])when δ > H p ( (cid:15)G sbd ) is O ( (cid:15) ) and oforder of magnitude (cid:15) in { χ ( p ) = 1 } ∩ {| x | ≥ µ/C } , by (8.6). Accepting aloss due to the positivity of ϑ , we choose µ so that hµ = (cid:15), µ = h(cid:15) (cid:29) . (8.26)Then, G (cid:15) ∈ S ( h + (cid:15) (cid:104) x (cid:105) ) , (8.27)and in particular, (cid:107) H G (cid:15) (cid:107) g = O (cid:18) h (cid:104) x (cid:105) + (cid:15) (cid:19) . (8.28)In addition to (8.25), (8.6), we know that H p G sbd ≥ . (8.29)Since χ ≺ χ , it follows that H p G (cid:15) (cid:38) χ ( p ) (cid:18) h (cid:104) x (cid:105) ϑ + (cid:15) {| x |≥ µ/C } (cid:19) (8.30)to be compared with the upper bound, that follows from (8.27) and the factthat supp G (cid:15) ⊂ p − ([ E − , E + ]): H p G (cid:15) = O (1) (cid:18) h (cid:104) x (cid:105) + (cid:15) (cid:19) , (8.31)where h (cid:104) x (cid:105) + (cid:15) (cid:16) h (cid:104) x (cid:105) + (cid:15) {| x |≥ µ/C } . (8.32) Remark 8.2
We define the spaces H (Λ υ(cid:15)G sbd ) , H (Λ υG (cid:15) ) as in (5.16) , with G replaced by υ(cid:15)G sbd and υG (cid:15) respectively. Since G lap ∈ S ( h ) by (8.23) , wesee that H υG (cid:15) − H υ(cid:15)G sbd = O ( υh ) , where H υG (cid:15) , H υ(cid:15)G sbd are defined in (5.2) ,with G replaced by υG (cid:15) and υ(cid:15)G sbd respectively. We conclude that (cid:107) u (cid:107) H (Λ υG(cid:15) ) (cid:16) (cid:107) u (cid:107) H (Λ υ(cid:15)G sbd ) , (8.33) uniformly with respect to υ , h , u , when ≤ υ ≤ υ (cid:28) , h ≤ h ≤ h (cid:28) . We now apply Proposition 5.5 and get (cid:61) ( P u | u ) H (Λ υG(cid:15) ) = (cid:90) Λ υG(cid:15) P top υ T u · T ue − υH (cid:15) /h dα + ( N υ u | u ) H (Λ υG(cid:15) ) , (8.34)71here we have preferred the more invariant integration on Λ υG (cid:15) rather thanto reduce everything to Λ . H (cid:15) is defined as in (5.2) with G replaced by G (cid:15) .Here (cf. (8.27)) P top υ = p top υ + O (1) υh ( h + (cid:15) (cid:104) x (cid:105) ) (cid:104) x (cid:105) = p top υ + O (1) υh (cid:104) x (cid:105) (cid:18) h (cid:104) x (cid:105) + (cid:15) (cid:19) , (8.35) p top υ ( ϑ ) = (cid:61) p ( (cid:60) ϑ + iυH G (cid:15) ( (cid:60) ϑ ))= − υH p G (cid:15) ( (cid:60) ϑ ) + O ( υ ) (cid:107) H G (cid:15) (cid:107) g = − υH p G (cid:15) ( (cid:60) ϑ ) + O ( υ ) (cid:18) h (cid:104) x (cid:105) + (cid:15) (cid:19) . (8.36)Further, N υ is negligible of order υ .Here we notice that by (8.32) h (cid:104) x (cid:105) ϑ + (cid:15) {| x |≥ µ/C } (cid:38) (cid:18) h (cid:104) x (cid:105) ϑ + (cid:15) ϑ (cid:19) =: m (cid:15) ( x ; h ) , (cid:15) ϑ := (cid:16) (cid:15)h (cid:17) ϑ (cid:15). (8.37)(In the limiting case, (cid:15) = h/C , where C > µ is of theorder of a large constant and (cid:15) ϑ (cid:16) h (cid:16) m (cid:15) .)Thus if we fix ϑ > υh (cid:104) x (cid:105) (cid:18) h (cid:104) x (cid:105) + (cid:15) (cid:19) , υ (cid:18) h (cid:104) x (cid:105) + (cid:15) (cid:19) (cid:28) υ (cid:18) h (cid:104) x (cid:105) ϑ + (cid:15) {| x |≥ µ/C } (cid:19) It then follows from (8.30), (8.35), (8.36) that − (cid:61) P top υ ( ρ ) ≥ υC χ ( p ) (cid:18) h (cid:104) x (cid:105) ϑ + (cid:15) {| x |≥ µ/C } (cid:19) − υ (cid:101) k υ , (8.38)where the right hand side is evaluated at the point (cid:60) ρ , (cid:101) k υ = O (1) (cid:18) h (cid:104) x (cid:105) + (cid:15) (cid:19) , supp (cid:101) k υ ⊂ p − ([ E − , E + ] \ [ E − δ , E + δ ]) . (8.39)We retain from this and (8.37), that − (cid:61) P top υ ≥ υC m (cid:15) ( x ; h ) − υk υ , (8.40)where k υ = O (cid:18) h (cid:104) x (cid:105) + (cid:15) (cid:19) , supp k υ ∩ p − ([ E − δ , E + δ ]) = ∅ . (8.41)72sing again the identification of h -pops and h -tops, we see that (cid:90) Λ υG(cid:15) υk υ T u · T ue − υH (cid:15) /h dα ≤ υ (cid:107) R υ u (cid:107) H (Λ υG(cid:15) ) + ( N (cid:48) υ u | u ) , (8.42)where N (cid:48) υ is negligible of order υ and R υ is an h -pop whose symbol is O (1) in S (( h/ (cid:104) x (cid:105) + (cid:15) ) / ) and with support disjoint from p − ([ E − δ , E + δ ]). Thuswith a new negligible operator of order υ , − (cid:61) ( P u | u ) H (Λ υG(cid:15) ) ≥ υC (cid:107) m / (cid:15) u (cid:107) H (Λ υG(cid:15) ) − υ (cid:107) R υ u (cid:107) H (Λ υG(cid:15) ) − ( N υ u | u ) . (8.43)Assume from now on that υ > P by P − z for z ∈ C , satisfying, (cid:60) z ∈ [ E − δ / , E + δ / , − C (cid:15) ϑ ≤ (cid:61) z ≤ C , (8.45)for
C > sufficiently large. Also, since the support of the symbol of R υ iscontained in a region where | p − z | ≥ / O (1), we have for every fixed N > (cid:107) R υ u (cid:107) H (Λ υG(cid:15) ) ≤ O (1) (cid:107) ( P − z ) u (cid:107) H (Λ υG(cid:15) ) + O (1) (cid:107) ( h/ (cid:104) x (cid:105) ) N ( h/ (cid:104) x (cid:105) + (cid:15) ) / u (cid:107) H (Λ υG(cid:15) ) . (8.46)Here (8.46) follows by the calculus of h -pseudodifferential operators associ-ated to Λ υG (cid:15) , see Section 6 of [16]. Using this in (8.43) with P there replacedby P − z , we get, − (cid:61) (( P − z ) u | u ) H (Λ υG(cid:15) ) ≥ C (cid:107) m / (cid:15) u (cid:107) H (Λ υG(cid:15) ) − C (cid:107) ( P − z ) u (cid:107) H (Λ υG(cid:15) ) . (8.47)Here, we have for every α > − (cid:61) (( P − z ) u | u ) H (Λ υG(cid:15) ) ≤ (cid:107) m (cid:15) ( x ; h ) − ( P − z ) u (cid:107) H (Λ υG(cid:15) ) (cid:107) m (cid:15) ( x ; h ) u (cid:107) H (Λ υG(cid:15) ) ≤ α (cid:107) m (cid:15) ( x ; h ) − ( P − z ) u (cid:107) H (Λ υG(cid:15) ) + 12 α (cid:107) m (cid:15) ( x ; h ) u (cid:107) H (Λ υG(cid:15) ) . Use this in (8.47) for a fixed large enough α together with the observation (cid:107) ( P − z ) u (cid:107) H (Λ υG(cid:15) ) (cid:46) (cid:107) m (cid:15) ( x ; h ) − ( P − z ) u (cid:107) H (Λ υG(cid:15) ) , to conclude that (cid:107) m (cid:15) ( x ; h ) u (cid:107) H (Λ υG(cid:15) ) ≤ O (1) (cid:107) m (cid:15) ( x ; h ) − ( P − z ) u (cid:107) H (Λ υG(cid:15) ) . (8.48)Summing up, we have: 73 roposition 8.3 Under the assumptions above, in particular (8.22) aboutnon trapping, we fix δ > small and then υ > small enough. Then for z in the region (8.45) , where C > is large enough, P − z : H (Λ υG (cid:15) , (cid:104) ξ (cid:105) ) → H (Λ υG (cid:15) ) is bijective and m (cid:15) ( x ; h ) ( z − P ) − m (cid:15) ( x ; h ) = O (1) : H (Λ υG (cid:15) ) → H (Λ υG (cid:15) ) , (8.49) where m (cid:15) is defined in (8.37) . Remark 8.4
By Remark , we can replace G (cid:15) with (cid:15)G sbd in (8.49) . Now G sbd vanishes for (cid:104) x (cid:105) ≤ µ/ (2 C ) and it follows that (cid:107) u (cid:107) H (Λ υG sbd ) (cid:16) (cid:107) u (cid:107) L for u with support in a fixed compact set. Moreover m (cid:15) ( x ; h ) (cid:16) h for x in anyfixed compact set. Hence from (8.49) , we deduce that χ ( z − P ) − χ = O (1 /h ) : L → L , for every fixed χ ∈ C ∞ ( R n ) . In order to shorten the notation, we will often write H sbd = H (Λ υ(cid:15)G sbd ) , D sbd = H (Λ υ(cid:15)G sbd , (cid:104) ξ (cid:105) ) , (8.50)where υ > E ∈ ] E − + δ, E + − δ [ be afixed energy level. (We can also allow it to vary, as we shall do in the nextsection, but then some geometric quantities will also vary.) Let ¨O (cid:98) R n bea connected open set (still assuming n = 1 but trying to keep the discussionas general as possible). Let U ⊂ ¨O be a compact subset. Assume: V − E < R n \ ¨O , V − E > \ U , V − E ≤ U , (8.51)diam d U = 0 , (8.52)where d is the Lithner-Agmon distance given by the metric ( V − E ) + ( x ) dx ,( V − E ) + = max( V − E, H p -flow has no trapped trajectories in p − ( E ) | R n \ ¨O . (8.53)Let M ⊂ ¨O be a connected compact set with smooth boundary such that M ⊃ { x ∈ ¨O; d ( x, ∂ ¨O) ≥ (cid:15) } , (8.54)74or some small (cid:15) >
0. Let P denote the Dirichlet realization of P in M ,equipped with the domain D ( P ) = H ( ◦ M ) ∩ H ( ◦ M ). (The right hand sidein (8.54) has smooth boundary, as we recalled after (1.36).)From Agmon estimates we have the well-known fact that if (cid:102) M ⊂ ¨Ohas the same properties as M with the same value of (cid:15) , then in any o (1)-neighborhood of E , the eigenvalues of P and (cid:101) P differ by O α (1) exp 2( α + (cid:15) − d ( U , ∂ ¨O)) /h for every α >
0. (Cf. [15])Let K ( h ) ⊂ C converge to { E } , when h → z ∈ K ( h ), z satisfies (8.45) , (8.55)dist ( z, σ ( P )) ≥ λ ( h ) , (8.56)where λ ( h ) > λ ( h ) ≥ − o (1) /h, h → . (8.57)Let (cid:101) V = V + W , where 0 ≤ W ∈ C ∞ (neigh ( U )) has its support in asmall neighborhood of U and V + W − E > (cid:101) P := − h ∆ + V + W , (cid:101) p ( x, ξ ) = ξ + V + W . Then (cid:101) p satisfies (8.22) if E − < E < E + and E + − E , E − E − are small enough. Then the resolvent estimate (8.49) applies to (cid:101) P for z ∈ K ( h ) and we recall Remark 8.4. We can then apply Agmon estimatesinside ¨O, as explained in Section 6 of [9], and we get( (cid:101) P − z ) − ( x, y ) = ˇ O ( e − d ( x,y ) /h ) , x, y ∈ ¨O , z ∈ K ( h ) , (8.58)where the notation ˇ O is explained in Proposition 9.3 in [16]. Under theassumptions (8.56), (8.57) we get the same estimate for P , i.e. we can replace( (cid:101) P , ¨O) by ( P , M ) in (8.58).Recall the elementary telescopic formula, for the moment under the apriori assumption that ( P − z ) − exists for z ∈ K ( h ) (which will follow fromthe discussion):( P − z ) − = ( (cid:101) P − z ) − + ( (cid:101) P − z ) − W ( (cid:101) P − z ) − + ( (cid:101) P − z ) − W ( P − z ) − W ( (cid:101) P − z ) − . (8.59)It reduces the study of ( P − z ) − to that of W ( P − z ) − W and we shallmake a perturbation series approach. Let χ ∈ C ∞ ( ◦ M ) be equal to 1 on { x ∈ ¨O; d ( x, ∂ ¨O) ≥ (cid:15) } and let χ ∈ C ∞ (neigh (supp W )) be equal to onenear supp W . Take the neighborhood small enough so that supp χ ∩ supp (1 − χ ) = ∅ . Let χ ∈ C ∞ ( ◦ M ) satisfy 1 B d ( U ,S / − (cid:15) ) ≺ χ ≺ B d ( U ,S / (cid:15) ) , where75 d ( U , r ) denotes the open ball of center U and radius r for the Lithner-Agmon distance and S := d ( U , ∂ ¨O). As a first approximation to ( P − z ) − ,we take E = χ ( P − z ) − χ + ( (cid:101) P − z ) − (1 − χ ) =: E + (cid:101) E. (8.60)Then,( P − z ) E = 1 + [ P, χ ]( P − z ) − χ − W ( (cid:101) P − z ) − (1 − χ ) =: 1 − K. (8.61)From (8.58) and the corresponding estimate for P , we see that (cid:107) K (cid:107) L ( m / (cid:15) H sbd ,L ( ¨O)) = (cid:101) O (1) exp (cid:18) − S h (cid:19) , (8.62)where (cid:101) O (1) indicates a quantity which is O ( e α/h ) for some α > (cid:15) and d (supp χ , U ) tend to 0. Thus for h > K + K + K + ... (8.63)converges to (1 − K ) − = 1 + (cid:101) O (1) exp (cid:0) − S h (cid:1) in L ( m / (cid:15) H sbd , L ( ¨O)). Itfollows that E (1 − K ) is a right inverse of P − z : D sbd → H sbd and since thelatter is of index 0 by the general theory of resonances ([16]) it is a two-sidedinverse. Proposition 8.5
Let C ⊃ K ( h ) → { E } , h → and assume (8.55) – (8.57) .Then for h > small enough and for z ∈ K ( h ) , P − z : D sbd → H sbd isbijective with inverse ( P − z ) − = χ ( P − z ) − χ (1 − K ) − + ( (cid:101) P − z ) − (1 − χ )(1 − K ) − . (8.64) Here, (cid:107) χ ( P − z ) − χ (1 − K ) − (cid:107) L ( H sbd , H sbd ) = O (1)dist ( z, σ ( P )) (8.65) and by Proposition , m (cid:15) ( x ; h ) ( (cid:101) P − z ) − (1 − χ )(1 − K ) − m (cid:15) ( x ; h ) = O (1) : H sbd → H sbd . (8.66)We next study the situation when z gets closer to σ ( P ). Let J ( h ) ⊂ R be an interval tending to { E } as h →
0. Assume that P has no spectrum in ∂J ( h ) + [ − δ ( h ) , δ ( h )] (8.67)76here the parameter δ ( h ) is small but not exponentially small;ln δ ( h ) ≥ − o (1) /h.σ ( P ) ∩ J ( h ) is a discrete set of the form { µ ( h ) , ..., µ m ( h ) } where m = m ( h ) = O ( h − n ) and we repeat the eigenvalues according to their multiplicity. LetΓ( h ) denote the set of resonances of P in J ( h ) − i [0 , (cid:15) ϑ /C ], C (cid:29)
1, alsorepeated according to their (algebraic) multiplicity. Assume that (cid:15) ≥ e − / ( Ch ) , (8.68)for some C (cid:29)
1, so that (cid:15) (cid:16) (cid:15)h (cid:17) ϑ ≥ e O ( h ) − S h . Then we have,
Proposition 8.6
There is a bijection b : { µ , ..., µ m } → Γ( h ) , such that b ( µ ) − µ = (cid:101) O ( e − S /h ) , where S = d ( U , ∂ ¨O) and the tilde indicates that the right hand side is O ( e ( ω − S ) /h ) , where ω = ω ( (cid:15) ) > and ω ( (cid:15) ) → , when (cid:15) → . We shall prove the proposition and also get precise information about theresolvent by studying an associated Grushin problem. Let e ( h ) , ..., e m ( h ) bean orthonormal system of eigenfunctions of P associated to the eigenvalues µ ( h ) , ..., µ m ( h ). Then we know from Chapter 6 of [9] that e j = ˇ O ( e − d ( U ,x ) /h ) . (8.69)A first trivial Grushin problem for P is defined by the matrix P = (cid:18) P − z R − R (cid:19) : D ( P ) × C m → L ( M ) × C m , (8.70)where R u ( j ) = ( u | e j ) , R − = ( R ) ∗ . (8.71)Let (cid:101) J ( h ) = J ( h ) + i [ − (cid:15) ϑ /C, /C ] (8.72)with C > z ∈ (cid:101) J ( h ), the operator P ( z ) is bijective with inverse E = (cid:18) E ( z ) E ( z ) E − ( z ) E − + (cid:19) : L ( M ) × C m → D ( P ) × C m , (8.73)77here, with Π denoting the spectral projection onto the space spanned by e , ..., e m , E ( z ) = ( P − z ) − (1 − Π ) = O (1 /δ ( h )) : L → D ( P ) , (8.74) E ( z ) v + = (cid:88) v + ( j ) e j , (cid:107) E (cid:107) L ( C m , D ( P )) ≤ O (1) , (8.75) E − ( z ) u ( j ) = ( u | e j ) , (cid:107) E − (cid:107) L ( L , C m ) ≤ O (1) , (8.76) E − + ( z ) = diag ( z − µ j ) . (8.77)Choose χ ∈ C ∞ ( ◦ M ) as after (8.59) and put R + = R χ : H sbd → C m , (8.78) R − = χR − : C m → H sbd . (8.79)Define, P ( z ) = (cid:18) P − z R − R + (cid:19) : D sbd × C m → H sbd × C m . (8.80)This is a Fredholm operator of index 0, so to show that it is bijective itsuffices to construct a right inverse.Let χ , χ , χ be as in the construction of ( P − z ) − in Proposition 8.5(where the assumptions on z were different). Following the same path asthere, we put (cid:101) E = (cid:18) χE χ χE E − χ E − + (cid:19) + (cid:18) ( (cid:101) P − z ) − (1 − χ ) 00 0 (cid:19) =: (cid:32) (cid:101) E (cid:101) E + (cid:101) E − (cid:101) E − + (cid:33) , (8.81) (cid:101) E = O (1 /δ ( h )) (cid:124) (cid:123)(cid:122) (cid:125) H sbd →D sbd + O (1) (cid:124) (cid:123)(cid:122) (cid:125) m / (cid:15) H sbd → m − / (cid:15) D sbd = O (1 /(cid:15) ϑ ) : H sbd → D sbd . (8.82)A straight forward calculation gives P ( z ) (cid:101) E ( z ) = (cid:18) A A A A (cid:19) , (8.83)where A = ( P − z ) χE χ + χR − E − χ + 1 − χ − W ( (cid:101) P − z ) − (1 − χ ) ,A = ( P − z ) χE + χR − E − + ,A = R χ E χ + R χ ( (cid:101) P − z ) − (1 − χ ) A = R χ E . E , ( (cid:101) P − z ) − , togetherwith the fact that E − χ = E − χ + (cid:101) O ( e − S / (2 h ) ) , we get( P − z ) χE χ + χR − E − χ = χ (( P − z ) E + R − E − (cid:124) (cid:123)(cid:122) (cid:125) =1 ) χ + (cid:101) O ( e − S / (2 h ) )= χ + (cid:101) O ( e − S / (2 h ) ) , and W ( (cid:101) P − z ) − (1 − χ ) = (cid:101) O ( e − S / (2 h ) ) . Hence A = 1 + (cid:101) O ( e − S / (2 h ) ) (8.84)Similarly, A = χ (( P − z ) E + R − E − + (cid:124) (cid:123)(cid:122) (cid:125) =0 ) + [ P, χ ] E (cid:124) (cid:123)(cid:122) (cid:125) = (cid:101) O ( e − S /h ) = (cid:101) O ( e − S /h ) , (8.85) A = R E (cid:124) (cid:123)(cid:122) (cid:125) =0 χ + R ( χ − M ) E χ (cid:124) (cid:123)(cid:122) (cid:125) = (cid:101) O ( e − S / (2 h ) ) + R χ ( (cid:101) P − z ) − (1 − χ ) (cid:124) (cid:123)(cid:122) (cid:125) = (cid:101) O ( e − S / (2 h ) ) = (cid:101) O ( e − S / (2 h ) ) , (8.86) A = R E (cid:124) (cid:123)(cid:122) (cid:125) =1 + R ( χ − M ) E = 1 + (cid:101) O ( e − S /h ) . (8.87)A first conclusion is that P ( z ) (cid:101) E ( z ) = 1 + (cid:101) O ( e − S / (2 h ) ) : H sbd × C m → H sbd × C m , where the remainder has entries with distribution kernels supported in M × M , M × C m , C m × M , ∅ respectively. so P ( z ) is bijective with inverse E ( z ) = (cid:101) E ( z )(1 + (cid:101) O ( e − S / (2 h ) )) = (cid:101) E ( z ) + (cid:101) O ( e − S / (2 h ) ) : m (cid:15) H sbd × C m → m − (cid:15) D sbd × C m (8.88)In particular, if we write E ( z ) = (cid:18) E ( z ) E + ( z ) E − ( z ) E − + ( z ) (cid:19) , (8.89)79e get E − + − E − + = (cid:101) O ( e − S / (2 h ) )and E satisfies the estimate in (8.82).We shall improve this estimate by working with exponential weights in¨O. For φ ∈ C ∞ ( ◦ M ) real-valued, we put H φ sbd = e φ/h H sbd equipped with the norm (cid:107) e − φ/h u (cid:107) H sbd . As a vector space it is equal to H sbd .The constructions above work without any great changes if we assume thatsupp ∇ φ ∩ U = ∅ , ( ∇ φ ) ≤ V − E − (cid:15) . By varying φ we see that( m (cid:15) Em (cid:15) )( x, y ) = ˇ O ( e − d ( x,y ) /h ) , ( m (cid:15) E + )( x ) = ˇ O ( e − d ( x,U ) /h ) , ( E − m (cid:15) )( y ) = ˇ O ( e − d ( U ,y ) /h ) , x, y ∈ B d ( U , S ) , (8.90)where we use the same symbols to denote the distribution kernels of E , E ± .For v + ∈ C m , the solution ( u, u − ) of the problem (cid:40) ( P − z ) u + R − u − = 0 ,R + u = v + , (8.91)is given by u = E + v + , u − = E − + v + . As an approximate solution to (8.91),we take u = χE v + , u − = E − + v + . Then (cid:40) ( P − z ) u + R − u − = [ P, χ ] E v + ,R + u = v + + R ( χ − M ) E v + , so we get the solution to (8.90) in the form (cid:40) u = u − E [ P, χ ] E v + − E + R ( χ − M ) E v + ,u − = u − − E − [ P, χ ] E v + − E − + R ( χ − M ) E v + . (8.92)Now it follows from (8.90) and the corresponding estimates for E , E ± , that | u − − u − | = (cid:101) O ( e − S /h ) | v + | , which means that E − + − E − + = (cid:101) O ( e − S /h ) . (8.93)80 roof of Proposition 8.6. By (8.77), (8.93) reads E − + − diag ( z − µ j ) = (cid:101) O ( e − S /h ) . (8.94)Now the resonances of P in (cid:101) J ( h ) are the zeros of det( E − + ) and we getthe proposition by means of elementary arguments for zeros of holomorphicfunctions of one variable. (cid:50) The first equation in (8.92) can be written E + v + − χE v + = − (cid:0) E [ P, χ ] E + E + R ( χ − M ) E (cid:1) v + and it follows that (cid:107) m (cid:15) ( E + v + − χE v + ) (cid:107) H sbd = (cid:101) O ( e − S /h ) | v + | , i.e. m (cid:15) ( E + − χE ) = (cid:101) O ( e − S /h ) : C m → H sbd . (8.95)Taking the adjoints with respect to the scalar product on L ( R n ) × C m ,we have P ( z ) ∗ E ( z ) ∗ = 1 , where P ( z ) ∗ = (cid:18) P ∗ − z R ∗ + R ∗− (cid:19) , E ( z ) ∗ = (cid:18) E ( z ) ∗ E − ( z ) ∗ E + ( z ) ∗ E − + ( z ) ∗ (cid:19) , and hence E ( z ) ∗ can be constructed by starting with (cid:98) E ( z ) = (cid:18) χE ∗ χ χE −∗ E ∗ χ E − + ∗ (cid:19) + (cid:18) ( (cid:101) P ∗ − z ) − (1 − χ ) 00 0 . (cid:19) In analogy with (8.95) we get m (cid:15) ( E ∗− − χE −∗ ) = (cid:101) O ( e − S /h ) : C m → H ∗ sbd , where we notice that H ∗ sbd = H (Λ − υG sbd ), in view of Proposition 8.8 of [16].By duality, ( E − − E − χ ) m (cid:15) = (cid:101) O ( e − S /h ) : H sbd → C m . (8.96)Recall the standard formula for Grushin problems:( z − P ) − = − E ( z ) + E + ( z ) E − + ( z ) − E − ( z ) , z ∈ (cid:101) J \ Γ( h ) . (8.97)81ere, E ( z ) is holomorphic and by (8.89), (8.81) , (8.88), E ( z ) = χE χ + ( (cid:101) P − z ) − (1 − χ ) + O ( e − S / (2 h ) ) : m (cid:15) H sbd → m − (cid:15) D sbd . (8.98) E ( z ) = O ( h/δ ) + O (1) : m (cid:15) H sbd → m − (cid:15) D sbd . (8.99)Here the term O ( h/δ ) represents the term χE χ which is O (1 /δ ) as anoperator on L ( R n ).When either m = 1 or dist ( z, σ ( P )) ≥ (cid:101) O ( e − S /h ), it follows from (8.93)that E − − + = O (cid:18) z, Γ( h )) (cid:19) . (8.100)Here we also assumed for simplicity that dist ( z, σ ( P ) ∩ J ( h )) = dist ( z, σ ( P ))which can be achieved by a slight shrinking of the interval J ( h ). Thus, when(8.100) holds, we get E + E − − + E − = O (cid:18) z, Γ( h )) (cid:19) : m (cid:15) H sbd → m − (cid:15) D sbd , (8.101)where we also used that by (8.95), (8.96), E + = O (1) : C m → m − (cid:15) D sbd , (8.102) E − = O (1) : m (cid:15) H sbd → C m . (8.103)Now (8.101) implies that ψE + E − − + E − ψ is O (1 / ( h dist ( z, Γ))) : L → L forevery ψ ∈ C ∞ . Using (8.95), (8.96) more directly, we get E + E − − + E − − χE E − − + E − χ = 1dist ( z, Γ) (cid:101) O ( e − S /h ) : m (cid:15) H sbd → m − (cid:15) D sbd (8.104)and here χE E − − + E − χ = O ( h/ dist ( z, Γ)) : m (cid:15) H sbd → m − (cid:15) D sbd . (8.105)From (8.97), (8.98), (8.99), (8.100), (8.102), (8.103), we get Proposition 8.7
We let z vary in the set (cid:101) J ( h ) in (8.72) . Assume that m = 1 or dist ( z, σ ( P )) ≥ (cid:101) O ( e − S /h ) , and also that dist ( z, σ ( P ) ∩ J ( h )) =dist ( z, σ ( P )) . Then we have, ( z − P ) − = O ( h/δ ) + O (1) + O ( h/ dist ( z, Γ)) : m (cid:15) H sbd → m − (cid:15) D sbd , (8.106) where the first two terms to the right are holomorphic in z . Back to adiabatics
Let I ⊂ R be an interval and let V t = V ( t, x ) ∈ C ∞ b ( I × R n ; R ) . (9.1)We assume that (cf. (8.3)) V t has a holomorphic extension (also denoted V t ) to { x ∈ C n ; |(cid:60) x | > C, |(cid:61) x | < |(cid:60) x | /C } such that V t ( x ) = o (1) , x → ∞ . (9.2) ∂ t V t ( x ) = 0 for | x | ≥ C, for some constant C > V ( t, x ) does not depend on h . However, when con-sidering a narrow potential wells in an island, of diameter (cid:16) h , we will have tomake an exception and allow such an h -dependence in a small neighborhoodof the well.Let 0 < E − < E (cid:48)− < E (cid:48) + < E + < ∞ and let E ( t ) ∈ C ∞ b ( I ; [ E (cid:48)− , E (cid:48) + ]) . (9.4)We assume that V t − E ( t ) has a potential well in an island as in Section 8.Let ¨O = ¨O( t ) (cid:98) R n be a connected open set and let U ( t ) ⊂ ¨O( t ) becompact. Assume (cf. (8.51)), V t − E ( t ) < R n \ ¨O( t ) ,> t ) \ U ( t ) , ≤ U ( t ) , (9.5)diam d t ( U ( t )) = 0 . (9.6)Here d t is the Lithner-Agmon distance on ¨O( t ), given by the metric ( V t − E ( t )) + dx .Also assume that with p t = ξ + V t ( x ),the H p t -flow has no trapped trajectories in p − t ( E ( t )) | R n \ ¨O( t ) . (9.7)It follows that d x V t (cid:54) = 0 on ∂ ¨O( t ) , (9.8)so ∂ ¨O( t ) is smooth and depends smoothly on t . Thus ¨O( t ) is a manifoldwith smooth boundary, depending smoothly on t . Further, U ( t ) depends83ontinuously on t . (This will still be true when we allow h -dependence near U ( t ).)For (cid:15) > M ( t ) = { x ∈ ¨O( t ); d t ( x, ∂ ¨O( t )) ≥ (cid:15) } , (9.9)so M ( t ) (cid:98) ¨O( t ) is a compact set with smooth boundary, depending smoothlyon t . (Here we use the structure of d t ( x, ¨O( t )) that follows from (9.8), see[16]).When I is a fixed compact interval, the assumptions above are fulfilleduniformly in t . Since we also want to allow I to be a very long interval, weadd the following compactness assumption:( V t , E ( t )) ∈ K , ∀ t ∈ I, where K is a compact subset of { V ∈ C ∞ b ( R n ; R ); V satisfies (9.2) with a fixed constant C } × [ E (cid:48)− , E (cid:48) + ]such that ( V, E ) satisfies the assumptions (9.3), with a fixed C as well as (9.5), (9.6), (9.7). (9.10)Let P ( t ) denote the Dirichlet realization of P ( t ) = − h ∆ + V t ( x ) on M ( t ). If we enumerate the eigenvalues of P ( t ) in ] E − , E + [ in increasing order(repeated with multiplicities) we know (as a general fact for 1-parameterfamilies of self-adjoint operators), that they are uniformly Lipschitz functionsof t . Let µ ( t ) = µ ( t ; h ) be such an eigenvalue and assume (cf. (8.67)), µ ( t ; h ) = E ( t ) + o (1) , h → , uniformly in t. (9.11) µ ( t ; h ) is a simple eigenvalue and σ ( P ( t )) ∩ [ E ( t ) − δ ( h ) , E ( t ) + δ ( h )] = { µ ( t ; h ) } . (9.12)Here, as in Section 8, δ ( h ) > δ ( h ) ≥ − o (1) /h, h → . (9.13)We restrict the spectral parameter z to D ( µ ( t ) , δ ( h ) / z − P ( t )) − = O (cid:18) | z − µ ( t ) | (cid:19) : L → D ( P ( t )) , (9.14)and more generally, ∂ kt ( z − P ( t )) − = O (cid:18) | z − µ ( t ) | k (cid:19) : L → D ( P ( t )) . (9.15)84trictly speaking, we work on sufficiently small time intervals, where we canreplace P ( t ) with the unitarily equivalent operator U ( t ) − P ( t ) U ( t ), where U ( t ) : L ( M ( t )) → L ( M ( t )) is induced by a diffeomorphism κ t : M ( t ) → M ( t ), depending smoothly on t . The spectral projection Π ( t ), associatedto ( P ( t ) , µ ( t )) is given byΠ ( t ) = 12 πi (cid:90) ∂D ( µ ( t ) ,r ) ( z − P ( t )) − dz, < r ≤ δ ( h ) / , (9.16)and choosing r = δ ( h ) /
2, we see that ∂ kt Π ( t ) = O ( δ ( h ) − k ) : L → D ( P ( t )) . (9.17)It follows that we can choose a normalized eigenfunction e ( t ): P ( t ) e ( t ) = µ ( t ) e ( t ) , (cid:107) e (cid:107) L = 1 , (9.18)such that ∂ kt e ( t ) = O ( δ ( h ) − k ) in D ( P ( t )) , k = 1 , , ..., (9.19)Now it is classical that µ ( t ) = ( P ( t ) e ( t ) | e ( t )) ,∂ t µ ( t ) = ( ∂ t P e | e ) + ( P ∂ t e | e ) + ( P e | ∂ t e ) , where the sum of the last two terms is equal to 0:( ∂ t e | P e ) + ( P e | ∂ t e ) = µ ( t )(( ∂ t e | e ) + ( e | ∂ t e )) = µ ( t ) ∂ t ( e | e ) = 0 . Thus, ∂ t µ ( t ) = ( ∂ t P e | e ) = O (1) , (9.20)and after differentiating in t : ∂ kt µ ( t ) = O ( δ ( h ) − k +1 ) , k = 1 , , ... For our purposes, it will be enough to work with the weaker estimate ∂ kt µ ( t ) = O ( δ ( h ) − k ) , k = 1 , , ... (9.21)Before discussing shape resonances, it will be convenient to discuss somesimple symmetry properties. In [16], (7.17) it was shown that( u | v ) H (Λ G ) = ( Bu | v ) L ( R n ) , u, v ∈ H (Λ G ) , B : H (Λ G ) → H (Λ − G ) is the sum of an elliptic Fourier integraloperator and a nop of order 1. (Here G denotes the function “ υG ” in Section8, where the parameter “ υ ’ is fixed according to (8.44).) Taking complexconjugates and exchanging u and v , we get( u | v ) H (Λ G ) = ( u | Bv ) L ( R n ) , u, v ∈ H (Λ G ) , . (9.22)Write (cid:104) u | v (cid:105) = (cid:90) R n uvdx (9.23)for the bilinear scalar product on L , so that (cid:104) u | v (cid:105) = ( u | Γ v ) L , where Γ v := v. Proposition 9.1
We have
Γ = O (1) : H (Λ G ) → H (Λ ˇ G ) , where ˇ G ( x, ξ ) := G ( x, − ξ ) . Proof.
This follows from 3 easily checked facts, where ˇ G ( x, ξ ) = G ( x, − ξ ):1) ( x, ξ ) ∈ Λ ˇ G ⇐⇒ ( x, − ξ ) ∈ Λ G .
2) If T is an FBI-transformation adapted to Λ ˇ G , then for u ∈ H (Λ G ), T Γ u ( α ) = (cid:101) T u ( α x , − α ξ ) , α ∈ Λ ˇ G , where (cid:101) T is an FBI-transformation adapted to Λ G .3) Let H be the function on Λ G , defined in (5.2) and let ˇ H be the corre-sponding function on Λ ˇ G . Thenˇ H ( x, ξ ) = H ( x, − ξ ) . (cid:50) Then, (cid:104) u | Γ Bv (cid:105) = ( u | v ) H (Λ G ) , u, v ∈ H (Λ G ) , (9.24)and here Γ B is an antilinear bijection H (Λ G ) → H (Λ − ˇ G ) with Γ B and(Γ B ) − uniformly bounded.Since P = P t is symmetric (with “t” indicating transpose for the bilinearscalar product), (cid:104) P u | v (cid:105) = (cid:104) u | P v (cid:105) , p ( x, − ξ ) = p ( x, ξ ) (also clear from the explicit formula p ( x, ξ ) = ξ + V ( x )), we see that − ˇ G is also an escape function and hence also G − ˇ G .Replacing G with the latter we get a new escape function G satisfying − ˇ G = G. (9.25)Now Γ B becomes an antilinear bijection: H (Λ G ) → H (Λ G ), uniformlybounded with its inverse. Replacing v with (Γ B ) − v in (9.24), we get (cid:104) u | v (cid:105) = ( u | (Γ B ) − v ) H (Λ G ) , u, v ∈ H (Λ G ) . (9.26)Then (cid:104) u | v (cid:105) is a bilinear nondegenerate scalar product on H (Λ G ) (In the caseof ordinary complex scaling, this is seen more directly by a shift of contourin (9.23).)We resume the earlier discussion with G = G sbd (assuming for simplicitythat the parameter “ υ ” in Section 8 is equal to 1) and apply Propositions8.6, 8.7 with J = [ µ ( t ; h ) − δ ( h ) / , µ ( t ; h ) + δ ( h ) / t ) := { z ∈ D ( µ ( t ; h ) , δ ( h ) / (cid:61) z ≥ − (cid:15) ϑ /C } , (9.27)where we recall that (cid:15) ϑ = ( (cid:15)/h ) ϑ (cid:15) . Then P ( t ) has a unique resonance λ ( t ) = λ ( t ; h ) in Ω( t ). It is simple and λ ( t ) − µ ( t ) = (cid:101) O ( e − S t /h ) , S t := d t ( U ( t ) , ∂ ¨O( t )) . (9.28)(8.106) gives( z − P ( t )) − = O (cid:18) hδ ( h ) (cid:19) + O (1) + O (cid:18) hz − λ ( t ) (cid:19) : m (cid:15) H sbd → m − (cid:15) D sbd , (9.29)where the first two terms in the right hand side are holomorphic in z . Inaddition to (8.24), (8.68) and the assumption ln δ ( h ) ≥ − o (1) /h , we assumefrom now on that (cid:15) ≤ δ ( h ) , (9.30)so that the first two terms in (9.29) drop out when | z − λ ( t ) | ≤ (cid:15) ϑ .We have the spectral projection π ( t ) = 12 πi (cid:90) ∂D ( λ ( t ) ,r ) ( z − P ( t )) − dz, (9.31)where 0 < r ≤ (cid:15) ϑ / (2 C ) and choosing the maximal value for r , we get from(9.29), π ( t ) = O ( h ) : m (cid:15) H sbd → m − (cid:15) D sbd . (9.32)87or the higher t derivatives, we write ∂ kt π ( t ) = 12 πi (cid:90) ∂D ( λ ( t ) ,r ) ∂ kt ( z − P ( t )) − dz, (9.33)where the integrand is a linear combination of terms,( z − P ( t )) − ( ∂ k t P )( z − P ( t )) − ( ∂ k t P ) ... ( ∂ k (cid:96) t P )( z − P ( t )) − , (9.34)with k j ≥ k + k + ... + k (cid:96) = k . In view of (8.37), we have m (cid:15) (cid:16) h onevery fixed compact set and since P is independent of t outside such a set,we conclude that ∂ k j t P ( t ) = O (1 /h ) : m − (cid:15) H sbd → m (cid:15) H sbd . Also, for z ∈ Ω( t ), we have( z − P ( t )) − = O ( h ) z − λ ( t ) : m (cid:15) H sbd → m − (cid:15) H sbd , when | z − λ ( t ) | (cid:46) (cid:15) ϑ . Hence the term (9.34) is O ( h )( z − λ ( t )) (cid:96) +1 : m (cid:15) H sbd → m − (cid:15) D sbd , so the integrand in (9.33) is O ( h )( z − λ ( t )) k +1 : m (cid:15) H sbd → m − (cid:15) D sbd , and we conclude that ∂ kt π ( t ) = O ( h ) (cid:15) kϑ : m (cid:15) H sbd → m − (cid:15) D sbd . (9.35)Let us fix t ∈ I for a while and write P = P ( t ). P is symmetric forthe bilinear scalar product (9.26) and so is the Grushin operator P in (8.80)( m = 1) if we use (cid:104) (cid:18) uu − (cid:19) | (cid:18) (cid:101) u (cid:101) u − (cid:19) (cid:105) = (cid:104) u | (cid:101) u (cid:105) + u − (cid:101) u − , and take care to use real eigenfunctions of P , when defining R ± . Then theinverse E : H sbd × C → D sbd × C is symmetric: E t = E, E t+ = E − . (9.36)88sing that ( z − P ) − = − E ( z ) + E + ( z ) E − + ( z ) − E − ( z )in (9.31), we get (with λ = λ ( t ) etc.), π = 1 E (cid:48)− + ( λ ) E + ( λ ) E − ( λ ) . (9.37)Here, by (8.94) and the Cauchy inequality, E (cid:48)− + = 1 + (cid:101) O (1) e − S t /h /(cid:15) ϑ . (9.38)From (cf. (8.68)), we get (cid:15) ϑ ≥ e − S t /h . (9.39)Now, by (8.95), (9.36) we have E + ( λ ) v + = v + e + , E − ( λ ) v = (cid:104) v | e + (cid:105) , (9.40) m (cid:15) ( e + − χe ) = (cid:101) O ( e − S t /h ) in D sbd . (9.41)Since m (cid:15) ≥ (cid:15) ϑ , this implies that, e + − χe = (cid:101) O ( e − S t /h /(cid:15) / ϑ ) in D sbd , (9.42) m − (cid:15) ( e + − χe ) = (cid:101) O ( e − S t /h /(cid:15) ϑ ) in D sbd . (9.43)From (9.37), (9.40), we get π u = 1 E (cid:48)− + ( λ ) (cid:104) u | e + (cid:105) e + , (9.44)and in particular that e + is a resonant state. The reproducing property π = π means that π e + = e + , which by (9.44) is equivalent to1 E (cid:48)− + ( λ ) (cid:104) e + | e + (cid:105) = 1 . (9.45)Put e = ( E (cid:48)− + ( λ )) − e + = (1 + (cid:101) O ( e − S t /h /(cid:15) ϑ )) e + . (9.46)Then π u = (cid:104) u | e (cid:105) e , (cid:104) e | e (cid:105) = 1 . (9.47)89e next estimate the t -derivatives of e = e ( t ). We work in a smallneighborhood of a variable point t ∈ I . Then f ( t ) = π ( t ) e ( t )is collinear to e ( t ) and we recover e ( t ) from the formula e ( t ) = (cid:104) f ( t ) | f ( t ) (cid:105) − f ( t ) . (9.48)By (9.35), we have (cid:107) m / (cid:15) ∂ kt f ( t ) (cid:107) D sbd = O ( h(cid:15) − kϑ ) (cid:107) m − / (cid:15) e ( t ) (cid:107) H sbd . (9.49)Using that m (cid:15) ≥ (cid:15) ϑ , we conclude that (cid:107) ∂ kt f ( t ) (cid:107) D sbd = O (1) h(cid:15) kϑ (cid:107) e (cid:107) H sbd = O (1) h(cid:15) k +1 ϑ , (9.50)for k ≥
1. For k = 0 we have (cid:107) f (cid:107) D sbd = 1 so we have the simpler but weakerestimate, (cid:107) ∂ kt f ( t ) (cid:107) D sbd = O (1) (cid:18) h(cid:15) ϑ (cid:19) k , k ≥ . (9.51)From (9.48) it then follows that (cid:107) ∂ kt e ( t ) (cid:107) D sbd = O (1) (cid:18) h(cid:15) ϑ (cid:19) k , k ≥ , (9.52)and this implies that ∂ kt π ( t ) = O (1) (cid:18) h(cid:15) ϑ (cid:19) k : H sbd → D sbd , k ≥ . (9.53)Next, we estimate ∂ k λ ( t ), k = 1 , , ... . We start with λ ( t ) = (cid:104) P ( t ) e ( t ) | e ( t ) (cid:105) . (9.54)By the symmetry of P ( t ) and the fact that (cid:104) e ( t ) | e ( t ) (cid:105) = 1, we get ∂ t λ ( t ) = (cid:104) ( ∂ t P ( t )) e ( t ) | e ( t ) (cid:105) = O (1) . (9.55)It follows from (9.52) that ∂ k +1 t λ = O (1) (cid:18) h(cid:15) ϑ (cid:19) k , k ≥ , ∂ kt λ = O (1) (cid:18) h(cid:15) ϑ (cid:19) ( k − + , k ≥ . (9.56)Recall from (8.24), (9.30), that (cid:15) ≤ min( h/C, δ ) , C (cid:29) . (9.57) λ does not depend on the choice of (cid:15) and if we make the maximal choice in(9.57), we get (cid:15) ϑ (cid:16) min(1 , δ/h ) ϑ min( h, δ ) and (9.56) gives ∂ kt λ = O (1) (cid:0) h min(1 , δ/h ) ϑ (cid:1) − ( k − + (9.58)Then (9.21) implies similar subexponential estimates for ∂ k ( λ − µ ), k ≥
1. Combining this with (9.28) and elementary interpolation estimates,we get ∂ kt ( λ − µ ) = (cid:101) O ( e − S t /h ) , k ≥ , (9.59)and we can then use (9.21) again, to get ∂ kt λ = O ( δ ( h ) − k ) , k ≥ . (9.60)We next study ( λ ( t ) − P ( t )) − (1 − π ( t )) and its derivatives. In thediscussion leading to (9.35) we have seen that m (cid:15) ∂ kt ( z − P ( t )) − m (cid:15) = O ( h )( z − λ ) k +1 : H sbd → D sbd , | z − λ | ≤ (cid:15) ϑ /C and hence ∂ kt ( z − P ( t )) − = O ( h ) (cid:15) ϑ ( z − λ ) k +1 : H sbd → D sbd , | z − λ | ≤ (cid:15) ϑ /C. Combining this with (9.53), we get ∂ kt (( z − P ( t )) − (1 − π ( t ))) = O (1) (cid:88) k + k = k h(cid:15) ϑ ( z − λ ) k +1 (cid:18) h(cid:15) ϑ (cid:19) k : H sbd → D sbd . When | z − λ | (cid:16) (cid:15) ϑ , the majorant is ≤ O (1) (cid:88) k + k = k h(cid:15) ϑ (cid:15) k ϑ (cid:18) h(cid:15) ϑ (cid:19) k ≤ O (1) (cid:88) k + k = k h(cid:15) ϑ (cid:18) h(cid:15) ϑ (cid:19) k (cid:18) h(cid:15) ϑ (cid:19) k ≤ O (1) (cid:18) h(cid:15) ϑ (cid:19) k +1 . z − P ( t )) − (1 − π ( t )) and its t -derivatives are holomorphic near z = λ ( t ) the maximum principle gives for | z − λ ( t ) | ≤ (cid:15) ϑ /C : ∂ kt (( z − P ( t )) − (1 − π ( t ))) = O (1) (cid:18) h(cid:15) ϑ (cid:19) k +1 : H sbd → D sbd . (9.61)With the Cauchy inequalities, this extends to ∂ (cid:96)z ∂ kt (( z − P ( t )) − (1 − π ( t ))) = O (1) (cid:15) − (cid:96)ϑ (cid:18) h(cid:15) ϑ (cid:19) k +1 = O (1) (cid:18) h(cid:15) ϑ (cid:19) k + (cid:96) +1 : H sbd → D sbd . (9.62)Finally we put z = λ ( t ) and get with the natural meaning of “lincomb” ∂ kt (cid:0) ( λ ( t ) − P ( t )) − (1 − π ( t )) (cid:1) = lincomb m + (cid:96) .. + (cid:96)λ = k,(cid:96)j ≥ ∂ mt ∂ λz (cid:0) ( z − P ( t )) − (1 − π ( t )) (cid:1) z = λ ( t ) ( ∂ (cid:96) t λ ) ... ( ∂ (cid:96) λ t λ )Using (9.62), (9.60), we see that the L ( H sbd , D sbd )-norm of the general termis O (1) (cid:15) − λϑ (cid:18) h(cid:15) ϑ (cid:19) m +1 δ − ( (cid:96) + ... + (cid:96) λ ) ≤ O (1) (cid:18) δ(cid:15) ϑ (cid:19) k − m (cid:18) h(cid:15) ϑ (cid:19) m +1 ≤O (1) h(cid:15) ϑ (cid:18) max( h, (cid:15) ϑ /δ ) (cid:15) ϑ (cid:19) k , where we used that λ ≤ (cid:96) + ...(cid:96) λ = k − m . Thus for every k ∈ N , ∂ kt (cid:0) ( λ ( t ) − P ( t )) − (1 − π ( t ) (cid:1) = O (1) h(cid:15) ϑ (cid:18) max( h, (cid:15) ϑ /δ ) (cid:15) ϑ (cid:19) k : H sbd → D sbd . (9.63)When (cid:15) is exponentially small, (cid:15) = exp( − / O ( h )), or more generally when (cid:15) ϑ ≤ δh , the estimate simplifies to ∂ kt (cid:0) ( λ ( t ) − P ( t )) − (1 − π ( t ) (cid:1) = O (1) (cid:18) h(cid:15) ϑ (cid:19) k +1 : H sbd → D sbd , k ≥ . (9.64)We next consider formal adiabatic solutions in the spirit of Proposition2.1. For the moment, we let (cid:15) , ε be independent parameters.92 roposition 9.2 Under the assumptions above, there exist two formal asy-mptotic series, ν ( t, ε ) ∼ ν ( t ) + εν ( t ) + ε ν ( t ) + ... in C ∞ ( I ; D sbd ) , (9.65) λ ( t, ε ) ∼ λ ( t ) + ελ ( t ) + ε λ ( t ) + ... in C ∞ ( I ) , (9.66) such that ( εD t + P ( t ) − λ ( t, ε )) ν ( t, ε ) ∼ as a formal asymptotic series in C ∞ ( I ; H sbd ) . Here, ∂ kt ν j = O (1)( h/ ˆ (cid:15) ϑ ) j + k in D sbd , j ≥ , k ≥ , (9.68) ∂ kt λ j = O (1)( h/ ˆ (cid:15) ϑ ) j − k , j ≥ , k ≥ . (9.69) Here, ˆ (cid:15) ϑ := (cid:15) ϑ max(1 , (cid:15) ϑ / ( δh )) / = min( (cid:15) ϑ , ( (cid:15) ϑ δh ) / ) . (9.70) Proof.
We sacrifice optimal sharpness for simplicity and work with theweaker form of (9.63): ∂ kt (cid:0) ( λ ( t ) − P ( t )) − (1 − π ( t ) (cid:1) = O (1)( h/ ˆ (cid:15) ϑ ) k , (9.71)which is equivalent to (9.64) in the most interesting case when (cid:15) ϑ ≤ δh . Weshall use (9.60): ∂ kt λ = O ( δ − k ) and the following weakened form of (9.52): ∂ kt e ( t ) = O (1)( h/ ˆ (cid:15) ϑ ) k in D sbd . (9.72)We follow the proof of Proposition 2.1 and annihilate successively thepowers of ε in the right hand side of (2.4). The first equation is then( P ( t ) − λ ( t )) ν ( t ) = 0 , (9.73)so we choose ν ( t ) = θ ( t ) e ( t ) (9.74)with the condition θ ( t ) (cid:16) , ∂ kt θ = O (1)( h/ ˆ (cid:15) ϑ ) k , (9.75)so that ν satisfies (9.68). Then the ε term in (2.4) vanishes.To annihilate the ε -term, we need to solve (2.9) which is solvable preciselywhen (cf. (2.10))0 = (cid:104) λ ( t ) ν ( t ) − D t ν ( t ) | e ( t ) (cid:105) = θ ( t ) λ ( t ) − (cid:104) D t ν ( t ) | e ( t ) (cid:105) . (cid:104) D t ν ( t ) | e ( t ) (cid:105) = D t θ ( t ) + θ ( t ) (cid:104) D t e ( t ) | e ( t ) (cid:105) = D t θ ( t ) , since (cid:104) D t e | e (cid:105) = 12 D t (cid:104) e | e (cid:105) = 0 , recalling that (cid:104) e | e (cid:105) = 1 . Thus, λ should satisfy θ ( t ) λ ( t ) − D t θ ( t ) = 0, λ ( t ) = D t θ θ , (9.76)and in particular, ∂ kt λ ( t ) = O (1)( h/ ˆ (cid:15) ϑ ) k , so λ satisfies (9.69). Remark 9.3
A natural choice of θ is θ = 1 . Then we get λ = 0 in (9.76) . With this unique choice of λ , we can solve (2.9) and the general solution is ν ( t ) = ( P ( t ) − λ ( t )) − (1 − π ( t ))( λ ( t ) ν − D t ν ( t )) + z ( t ) e ( t ) , (9.77)where we are free to choose z ( t ), and we will take z ( t ) = 0 for simplicity.From (9.71), the estimate (9.69) for λ and (9.68) for ν , we get ∂ kt ν = O (1)( h/ ˆ (cid:15) ϑ ) k in D sbd , k ≥ , (9.78)i.e. ν satisfies (9.68).The equation for annihilating the ε j -term in (2.4) is( P ( t ) − λ ( t )) ν j = ( λ − D t ) ν j − + λ ν j − + ... + λ j − ν + λ j ν . (9.79)Let N ≥ ν j , λ j for j ≤ N − j ≤ N −
1. Consider (9.79) for j = N . Thecondition for finding a solution ν N is that the right hand side is orthogonal(for (cid:104)·|·(cid:105) ) to e and since (cid:104) ν | e (cid:105) = θ ( t ), we get λ N = θ − (cid:104) ( D t − λ ) ν N − + λ ν N − + ... + λ N − ν | e (cid:105) . (9.80)Here ∂ kt D t ν N − = O (1)( h/ ˆ (cid:15) ϑ ) N − k +1 = O (1)( h/ ˆ (cid:15) ϑ ) N − k (9.81)and for 1 ≤ (cid:96) ≤ N − ∂ kt ( λ (cid:96) ν N − (cid:96) ) = O (1)( h/ ˆ (cid:15) ϑ ) (cid:96) − N − (cid:96) )+ k = O (1)( h/ ˆ (cid:15) ϑ ) N − k . (9.82)Using also (9.75), we see that λ N satisfies (9.69).94e can now solve for ν N in (9.79): ν N = ( P ( t ) − λ ( t )) − (1 − π ( t ))(( λ − D t ) ν N − + λ ν N − + ... + λ N − ν + λ N ν ) + z ( t ) e ( t ) . (9.83)Again we take z = 0 for simplicity and get, using (9.71), (9.81), (9.82): ∂ kt ν N = O (1)( h/ ˆ (cid:15) ϑ ) N − k = O (1)( h/ ˆ (cid:15) ϑ ) N + k , so ν N satisfies (9.68) and this finishes the inductive proof. (cid:50) Remark 9.4
The construction of ν j , λ j is independent of the choice of ambi-ent spaces and if we choose (cid:15) maximal in (9.57) we see as after that inequalitythat (9.69) becomes ∂ kt λ j = O (1) (cid:0) min(1 , δ/h ) ϑ min( δ, min(1 , δ/h ) ϑ ) (cid:1) − (2 j + k − , j ≥ , k ≥ . (9.84) When δ ≤ h this simplifies to ∂ kt λ j = O (1) (cid:32)(cid:18) hδ (cid:19) ϑ δ (cid:33) j − k . This can probably be improved as in the proof of (9.60) . We continue the discussion under the assumptions of Proposition 9.2. Putfor N ≥ ν ( N ) = ν + εν + ... + ε N ν N , (9.85) λ ( N ) = λ + ελ + ... + ε N λ N , N ≥ . (9.86)Then by construction (cf. (2.4)),( εD t + P ( t ) − λ ( N ) ) ν ( N ) = r ( N +1) , (9.87)where r ( N +1) = ε N +1 D t ν N − (cid:88) j,k ≤ Nj + k ≥ N +1 ε j + k λ j ν k . (9.88)From the estimates in Proposition 9.2, we get r ( N +1) = O (1) ε N +1 ( h/ ˆ (cid:15) ϑ ) N +1 + (cid:88) j,k ≤ Nj + k ≥ N +1 ε j + k ( h/ ˆ (cid:15) ϑ ) j + k ) − in D sbd .
95n the following, we assume that ε h ˆ (cid:15) ϑ (cid:28) . (9.89)Recall from (9.13) that δ = δ ( h ) is small, but not exponentially small andthat (cid:15) ϑ = ( (cid:15)/h ) ϑ (cid:15) . Then (9.89) holds if we assume that ε is exponentiallysmall: 0 < ε ≤ O (1) exp ( − / ( Ch )) , for some C > , (9.90)and choose (cid:15) ≥ ε ϑ ) − α , (9.91)for some α ∈ ]0 , / (4(1 + ϑ ))[.Having assumed (9.89) we get r ( N +1) = O (1) ε (cid:16) ε h ˆ (cid:15) ϑ (cid:17) N +1 in D sbd andmore generally, ∂ kt r ( N +1) = O (1) ε (cid:32) ε h ˆ (cid:15) ϑ (cid:33) N +1 (cid:18) h ˆ (cid:15) ϑ (cid:19) k in D sbd . (9.92)Also, since (cid:107) ν ( t ) (cid:107) H sbd = (cid:107) e ( t ) (cid:107) H sbd , we get (cid:107) ν ( N ) ( t ) (cid:107) H sbd = (1 + O ( εh / ˆ (cid:15) ϑ )) (cid:107) ν ( t ) (cid:107) H sbd (cid:16) . (9.93)Recall (7.30) with the subsequent observation and the choice of µ in(8.26): − (cid:61) ( P ( t ) u | u ) H sbd ≥ −O ( (cid:15) ∞ ) (cid:107) u (cid:107) H sbd . (9.94)Let I (cid:51) t (cid:55)→ u ( t ) ∈ H (Λ (cid:15)G , (cid:104) ξ (cid:105) ) be continuous such that ∂ t u is continuouswith values in H (Λ (cid:15)G , (cid:104) ξ (cid:105) − ), G = G sbd . Assume that u is a solution of( εD t + P ( t )) u ( t ) = 0 . Then, ε∂ t (cid:107) u ( t ) (cid:107) H sbd = 2 (cid:61) ( P ( t ) u | u ) ≤ O ( (cid:15) ∞ ) (cid:107) u (cid:107) H sbd , implying (cid:107) u ( t ) (cid:107) H sbd ≤ e O ( (cid:15) ∞ )( t − s ) /ε (cid:107) u ( s ) (cid:107) H sbd , t ≥ s. Assume (cid:15) ≤ O ( ε /N ) , for some fixed N > . (9.95)Then, (cid:107) u ( t ) (cid:107) H sbd ≤ e O ( (cid:15) ∞ )( t − s ) (cid:107) u ( s ) (cid:107) H sbd , t ≥ s. (9.96)96rom (9.94)and the fact that P ( t ) − z : D sbd → H sbd is Fredholm ofindex 0, when (cid:61) z >
0, we see that (cid:107) ( P ( t ) − z ) − (cid:107) L ( H sbd , H sbd ) ≤ (cid:61) z − y (cid:15) , for (cid:61) z > y (cid:15) , (9.97)where y (cid:15) = O ( (cid:15) ∞ ) can be chosen independent of t ∈ I . By the Hille-Yosidatheorem, − iP ( t ) generates a strongly continuous semi-group leading to: If u ∈ D sbd , then ∃ ! u ∈ C ([0 , + ∞ [; D sbd ) ∩ C ([0 , + ∞ [; H sbd ) such that( εD t + P ( t )) u ( t ) = 0 for t ≥ , u (0) = u , (9.98)where we entered the parameter ε > P ( t ) − P ( t ) is a smooth function of t with values in L ( H sbd , H sbd ) andan application of [22, Theorem 6.1 and Remark 6.2], allows us to concludethat for every u ∈ D sbd and every s ∈ I , there exist u ∈ C ( I ∩ [ s, ∞ [; D sbd ) ∩ C ( I ∩ [ s, ∞ [; H sbd ) such that( εD t + P ( t )) u ( t ) = 0 for s ≤ t ∈ I, u ( s ) = u . (9.99)Again the solution satisfies (9.96).This allows us to define the forward fundamental matrix E ( t, s ), I (cid:51) t ≥ s ∈ I of εD t + P ( t ): (cid:40) ( εD t + P ( t )) E ( t, s ) = 0 , t ≥ s,E ( t, t ) = 1and from [22, Theorem 6.1 and Remark 6.2] we infer, in particular, that E ( t, s ) is strongly continuous in the H sbd -norm both in t and s , such that (cid:107) E ( t, s ) (cid:107) L ( H sbd , H sbd ) ≤ exp(( t − s ) O ( (cid:15) ∞ /ε )) , t ≥ s, t, s ∈ I. (9.100)If v ∈ C ( I ; H sbd ) vanishes for t near inf I , we can solve ( εD t + P ( t )) u = v on I by u ( t ) = iε (cid:90) t inf I E ( t, s ) v ( s ) ds. Now return to (9.85)–(9.87) with λ j , ν j as in Proposition 9.2 and r ( N +1) satisfying (9.92). We notice that λ ( N ) = λ + O (1) ε ε h ˆ (cid:15) ϑ , (9.101)97nd that this improves to λ ( N ) = λ + O (1) ε (cid:32) ε h ˆ (cid:15) ϑ (cid:33) , (9.102)if we take θ = 1 , λ = 0 (9.103)as in Remark 9.3. We assume (9.103) in the following.Assume, to fix the ideas, that 0 ∈ I , and restrict the attention to I + = { t ∈ I ; t ≥ } . From (9.87), we get( εD t + P ( t )) u ( N ) = ρ ( N +1) , t ∈ I + , (9.104)where u ( N ) = e − i (cid:82) t λ ( N ) ds/ε ν ( N ) , ρ ( N +1) = e − i (cid:82) t λ ( N ) ds/ε r ( N +1) . (9.105)By (9.93), (9.92), we have (cid:107) ρ ( N +1) (cid:107) H sbd = O (1) ε (cid:32) ε h ˆ (cid:15) ϑ (cid:33) N +1 (cid:107) u ( N ) (cid:107) H sbd . (9.106)Taking the imaginary part of the scalar product in H sbd with u ( N ) , we getwith norms and scalar products in H sbd : − ε∂ t (cid:107) u ( N ) (cid:107) + ( (cid:61) P u ( N ) | u ( N ) ) = (cid:61) ( ρ ( N +1) | u ( N ) ) ,ε∂ t (cid:107) u ( N ) (cid:107) = 2( (cid:61) P u ( N ) | u ( N ) ) − (cid:61) ( ρ ( N +1) | u ( N ) ) ≤ O ( (cid:15) ∞ ) (cid:107) u ( N ) (cid:107) + 2 (cid:107) ρ ( N +1) (cid:107)(cid:107) u ( N ) (cid:107) . Hence, by (9.106) and the assumption (9.95), ε∂ t (cid:107) u ( N ) (cid:107) ≤ O (1) ε (cid:32) ε h ˆ (cid:15) ϑ (cid:33) N +1 (cid:107) u ( N ) (cid:107) , leading to (cid:107) u ( N ) ( t ) (cid:107) ≤ e O (1) tε − / ( ε / h/ ˆ (cid:15) ϑ ) N +1 (cid:107) u ( N ) (0) (cid:107) , ≤ t ∈ I. (9.107)Assume, (sup I ) ε − (cid:32) ε h ˆ (cid:15) ϑ (cid:33) N +1 ≤ O (1) . (9.108)98hen, for 0 ≤ t ∈ I , (cid:107) u ( N ) ( t ) (cid:107) H sbd ≤ O (1) (cid:107) u ( N ) (0) (cid:107) H sbd , (cid:107) ρ ( N +1) ( t ) (cid:107) H sbd ≤ O (1) ε (cid:32) ε h ˆ (cid:15) ϑ (cid:33) N +1 (cid:107) u ( N ) (0) (cid:107) H sbd . (9.109)Using the fundamental matrix E ( t, s ) to correct the error ρ ( N +1) we havethe exact solution u = u ( N )exact , u = u ( N ) − iε (cid:90) t E ( t, s ) ρ ( N +1) ( s ) ds (9.110)of the equation ( εD t + P ( t )) u = 0 on I + . From (9.108) we get sup I ≤ ε − N , (9.111)for some fixed finite N . Then by (9.100), (9.95), (cid:107) E ( t, s ) (cid:107) L ( H sbd , H sbd ) ≤ e O ( ε ∞ ) = 1 + O ( ε ∞ ) , (9.112)and using this and (9.109) in (9.110), we get (cid:107) u − u ( N ) (cid:107) H sbd ≤ O (1) ε − (sup I ) ε (cid:32) ε h ˆ (cid:15) ϑ (cid:33) N +1 (cid:107) u ( N ) (0) (cid:107) H sbd . (9.113)This estimate is the main result of the present work. Let us recollect theassumptions and the general context in the following theorem. Theorem 9.5
Let V t = V ( t, x ) ∈ C ∞ b ( I × R n ; R ) , where n = 1 , < E −
0. Put (cid:98) V t = (cid:101)(cid:98) V t − α t U . Then (cid:98) P t := − h ∆+ (cid:98) V t is a self-adjoint operator (defined by means of Friedrichs extension)with purely discrete spectrum in ] − ∞ , V ( x ) − δ [, bounded from below bymin( (cid:101) V t ( x ) − δ , (cid:101) V t ( x ) − α t − O ( h )).The eigenvalues in the interval ] − ∞ , V ( x ) − δ [ can be obtained byscaling and simple semi-classical analysis: Let e ( α ) < e ( α ) < ... < e k ( α ) ( α ) < − ∂ − α ] − , ( x ) on R . Then for h < (cid:98) P t in ] − ∞ , V ( x ) − δ [ are of the form (cid:98) E k ( t ; h ) ∼ E k, ( t ) + hE k, ( t ) + ... (9.123)where E k, ( t ) = (cid:101) V t ( x ) + e k ( α t ) (9.124)belongs to ] − ∞ , (cid:101) V t ( x ) − δ /
2[ (in the limit of small h ) and we get all sucheigenvalues this way (one for each k ) when h > k ∈ N and assume that we have the well-defined eigenvalue (cid:98) E k ( t ; h ) =: (cid:98) E ( t ; h ) of (cid:98) P t in ] δ , (cid:101) V t ( x ) − δ [ for all t ∈ I for 0 < h (cid:28)
1. (Thepositivity is required since we look for shape resonances of P t .) Define the t -dependent potential island¨O( t ) = { x ∈ R n ; (cid:101) V t ( x ) > E ( t ) } , E ( t ) := E k, ( t ) . (9.125)Let (cid:101) p t = ξ + (cid:101) V t ( x ) be the semi-classical principal symbol of (cid:101) P t . Assumethat The H (cid:101) p t -flow is non-trapping on ( (cid:101) p t ) − ( E ( t )) | R n \ ¨O . (9.126)In ¨O( t ) we have the Lithner-Agmon distance d t , associated to the metric( (cid:101) V t ( x ) − E ) dx . Let S t := d t ( x , ∂ ¨O( t )) > M ( t ) := { x ∈ ¨O( t ); (cid:101) V t ( x ) > E ( t ) + δ } , where δ > M ( t ) has smooth boundary anddepends smoothly on t . Let P t be the Dirichlet realization of P t in M ( t ).Then P t has a unique eigenvalue E ( t ; h ) such that E ( t ; h ) = (cid:98) E ( t ; h ) = o (1), h → E ( t ; h ) = (cid:98) E ( t ; h ) + O ( e − / ( Ch ) ) . (9.127)101s in the beginning of this section 8 we know that P t has a unique resonancewith (cid:60) λ ( t ) − E ( t ; h ) = o (1) , (cid:61) λ ( t ) ≥ − Ch ln(1 /h ) . Moreover, we have λ ( t ; h ) = E ( t ; h ) + (cid:101) O ( e − S t /h ) . (9.128)This means that (apart from the fact that our potential is h -dependentnear U ) we can apply Theorem 9.5 with δ ( h ) (cid:16) References [1] W. K. Abou Salem and J. Fr¨ohlich,
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