An example of resonance instability
Jean-Francois Bony, Setsuro Fujiie, Thierry Ramond, Maher Zerzeri
aa r X i v : . [ m a t h . SP ] M a y AN EXAMPLE OF RESONANCE INSTABILITY
JEAN-FRANC¸ OIS BONY, SETSURO FUJII´E, THIERRY RAMOND, AND MAHER ZERZERI
Abstract.
We construct a semiclassical Schr¨odinger operator such that the imaginary partof its resonances closest to the real axis changes by a term of size h when a real compactlysupported potential of size o ( h ) is added. Introduction
In this note, we consider semiclassical Schr¨odinger operators P on L ( R n ), n ≥ P = − h ∆ + V ( x ) , where V ∈ C ∞ ( R n ; R ) is a real-valued smooth compactly supported potential. Depending onthe situation, one may also work with such operators outside a compact smooth obstacle withDirichlet boundary condition. Since P is a compactly supported perturbation of − h ∆, theresonances of P near the real axis are well-defined through the analytic distortion method orusing the meromorphic extension of its truncated resolvent. We send back the reader to thebooks of Sj¨ostrand [14] or Dyatlov and Zworski [6] for a general presentation of resonancetheory, and we denote Res( P ) the set of resonances of P .The stability of the resonances is a rather touchy question. Indeed, we do not knowyet whether the concept of resonance persists under the perturbation by a non-analyticnon-exponentially decreasing potential. Therefore, we only consider here perturbations ofSchr¨odinger operators (1.1) by subprincipal real-valued smooth compactly supported poten-tials of the form h τ W ( x ) with τ > W ∈ C ∞ ( R n ; R ). But even in this setting, thestability of resonances is a subtle problem since stability results and instability results can beobtained for the same operator.On one hand, the resonances tend to be stable as other spectral objects like the eigenval-ues. This is particularly clear when the resonances are defined by complex distortion, sincethe usual perturbation theory of discrete spectrum can be directly applied to the distortedoperator. But, even if the resonances are defined as the poles of the meromorphic extensionof some weighted resolvent, Agmon [1, 2] has proved their stability. On the other hand, theresonances can be unstable since they do not come from a self-adjoint problem. Thus, sometypical non self-adjoint effects may occur concerning the resonances even if P is self-adjoint.For instance, the distorted operator may have a Jordan block or the truncated resolvent mayhave a pole of algebraic order greater than 1 (see e.g. Sj¨ostrand [13, Section 4]).Our instability result is the following. Mathematics Subject Classification.
Key words and phrases.
Resonances, semiclassical asymptotics, microlocal analysis, spectral instability,Schr¨odinger operators.
Acknowledgments:
The second author was partially supported by the JSPS KAKENHI Grant 18K03384. − D h − αh Ah Bh − D h − δh − Ch Ch z − E Res (cid:0) P + h δ W (cid:1) Res( P ) Figure 1.
The spectral setting of Theorem 1.1.
Theorem 1.1 (Resonance instability) . In dimension n = 2 , one can construct an operator P and a potential W as above satisfying the following property for all δ > small enough.There exist a set H ⊂ ]0 , with ∈ H and constants D , E , α > such that, for all C > and − C ≤ A < B ≤ C , i ) On one hand, P has no resonance z with Re z ∈ E + [ − Ch, Ch ] and (1.2) Im z ≥ − D h − αh, for h ∈ H small enough. ii ) On the other hand, the resonances z of P + h δ W with Re z ∈ E + [ Ah, Bh ] closestto the real axis satisfy (1.3) Im z ∼ − D h − δh, for h ∈ H small enough. The result is illustrated in Figure 1. Theorem 1.1 ii ) provides at least one resonance z of P + h δ W satisfying Re z ∈ E + [ Ah, Bh ] and Im z ∼ − D h − δh . But its proof shows thatthe number of such resonances is at least of order | ln h | . In particular, the essential quantumtrapping in E + [ Ah, Bh ] defined by(1.4) ess-qt( Q ) = lim n → + ∞ lim sup h → h ∈ H inf z ,...,z n ∈ Res( Q )Re z • ∈ E +[ Ah,Bh ] sup z ∈ Res( Q ) \{ z ,...,z n } Re z ∈ E +[ Ah,Bh ] h | Im z | , increases by at least ( α − δ )( D + α ) − ( D + δ ) − when we add the perturbation h δ W tothe operator P . Thus, the resonance instability described here is not an anomaly due to anexceptional resonance or a Jordan block but a phenomenon mixing geometry and analysis.In the statement of the previous result, we do not specify the subset of semiclassical param-eters H . In fact, depending on the geometric situation, the resonance instability may occuron the whole interval H =]0 ,
1] or only near a sequence H like { j − ; j ∈ N ∗ } . Operatorscorresponding to these different situations are given at the end of Section 2.For 0 < κ ≪ z of P + κhW with Re z ∈ E + [ Ah, Bh ] closest to the real axis satisfy Im z ∼ − D h for h ∈ H small enough. Theproof of this point is similar to that of Theorem 1.1. On the contrary, for larger values of κ , some cancellations may appear and P + κhW may have a resonance free region of size D h + αh below the real axis as for P . N EXAMPLE OF RESONANCE INSTABILITY 3 E ( C ) E E ( B )( A ) Figure 2.
The resonances generated by ( A ) a well in the island, ( B ) a non-degenerate critical point and ( C ) a hyperbolic closed trajectory.The constructions in Theorem 1.1 can be realized in any dimension n ≥
2, but our methodof proof does not work in dimension n = 1. Indeed, the Hamiltonian vector field must havean anisotropic hyperbolic fixed point. Nevertheless, we do not know yet if the resonanceinstability phenomenon described here occurs in dimension one.Let P θ denote the operator P in the proof of Theorem 1.1 after a complex distortion ofangle θ = h | ln h | . Its resolvent satisfies a polynomial estimate in Ω = E + [ − Ch, Ch ] + i [ − D h − αh, h ]. This means that, for some M >
0, we have(1.5) (cid:13)(cid:13) ( P θ − z ) − (cid:13)(cid:13) . h − M , uniformly for z ∈ Ω. By the usual perturbation argument, it implies that P + Q has noresonance in Ω for any distortable perturbation Q of size o ( h M ). The stability of resonancesunder small enough perturbations has already been observed (see e.g. Agmon [1, 2]). Sum-ming up, the resonances of P are stable for perturbations of size o ( h M ) and unstable for someperturbations of size h δ (showing that M ≥ δ ).The present result is obtained for a Schr¨odinger operator whose trapped set at energy E consists of a hyperbolic fixed point and homoclinic trajectories, following our recent paper[4]. In fact, the instability phenomenon obtained here does not hold in the geometric settingspreviously studied (see Figure 2). In the “well in the island” situation, the resonances areknown to be exponentially close to the real axis (see Helffer and Sj¨ostrand [10] for globallyanalytic potentials and Lahmar-Benbernou, Martinez and the second author [7] for poten-tials analytic at infinity). Adding a subprincipal real potential hW ( x ) does not change thisproperties. When the trapped set at energy E consists of a non-degenerate critical point(say at ( x , ∈ T ∗ R n ), Sj¨ostrand [13] has proved that the resonances form, modulo o ( h ), aquarter of a rectangular lattice which is translated by hW ( x ) when a subprincipal potential hW ( x ) is added. Finally, the asymptotic of the resonances generated by a hyperbolic closedtrajectory has been obtained by G´erard and Sj¨ostrand [9] (see also Ikawa [11] and G´erard [8]for obstacles). Modulo o ( h ), they form half of a rectangular lattice which is translated by areal quantity after perturbation by a real potential hW ( x ). Summing up, the imaginary partof the resonances is very stable in the three previous examples: it moves only by o ( h ) when aperturbation by a real potential of size h is applied. In other words, if the quantum trapping (or maximum of the quantum lifetime) in E + [ − Ch, Ch ] of an operator Q is defined by(1.6) qt( Q ) = lim sup h → sup z ∈ Res( Q )Re z ∈ E +[ − Ch,Ch ] h | Im z | , J.-F. BONY, S. FUJII´E, T. RAMOND, AND M. ZERZERI with the conventions that qt( Q ) = + ∞ if the limit diverges and qt( Q ) = 0 if Q has noresonance, we have qt( P ) = qt( P + hW ) in these examples. The situation is completelyopposite in Theorem 1.1 since a self-adjoint perturbation of size o ( h ) induces a change ofsize 1 of the quantum trapping. By definition, we always have qt( Q ) ∈ [0 , + ∞ ] and qt( Q ) ≥ ess-qt( Q ). Moreover, if the resonance expansion of the quantum propagator holds, we have k χe − itQ/h ϕ ( Q ) χ k ≈ e t/ qt( Q ) for t ≫ h in an appropriate sequence, justifying the nameof quantum trapping. Other results in scattering theory provide resonance free regions, thatis upper bounds on the quantum trapping, under geometric assumptions. In general, thebounds obtained do not depend on the subprincipal symbol, assumed to be self-adjoint in anappropriate class (see for instance Section 3.2 of Nonnenmacher and Zworski [12]). In thepresent setting, Section 3.1 of [4] implies qt( P ) ≤ D − , but Theorem 1.1 i ) shows that thisinequality is not sharp.Theorem 1.1 may seem natural since the distorted resolvent is generally large in the un-physical sheet and small perturbations may produce eigenvalues. More precisely, the norm ofthe distorted resolvent is known to be larger than h − , that is (cid:13)(cid:13) ( P θ − z ) − (cid:13)(cid:13) ≫ h − , with Im z <
0, in many cases (see e.g. Burq and two of the authors [3] or Dyatlov and Waters[5]). By the pseudospectral theory (see e.g. Section I.4 of Trefethen and Embree [15]), thereexists a bounded operator Q θ of size o ( h ) such that z is precisely an eigenvalue of P θ + Q θ .Nevertheless, it is not clear that Q θ is the distortion of some operator Q , that Q is a potentialand that Q is self-adjoint. In fact, as explained in the previous paragraph, this is not alwaysthe case.This instability phenomenon is due to the non self-adjoint nature of the resonances (evenfor self-adjoint operators). Such a property never holds for the usual spectrum in the self-adjoint framework. Indeed, for any self-adjoint operator P and any bounded perturbation W , the spectrum of P + W satisfies σ ( P + W ) ⊂ σ ( P ) + B (0 , k W k ) . Thus, a perturbation of size h δ of a self-adjoint operator can not lead to a perturbation ofsize h of its spectrum.The operator P and the potential W are constructed in Section 2. The instability phe-nomenon stated in Theorem 1.1 is proved in Section 3.2. Construction of the operators
To construct a Schr¨odinger operator P = − h ∆+ V ( x ) as in (1.1) with unstable resonances,we follow Example 4.23 and Example 4.24 (B) of [4]. We send back the reader to this paperfor a slightly different presentation, some close geometric situations and general results aboutresonances generated by homoclinic trajectories. As usual, p ( x, ξ ) = ξ + V ( x ) denotes thesymbol of P , its associated Hamiltonian vector field is H p = ∂ ξ p · ∂ x − ∂ x p · ∂ ξ = 2 ξ · ∂ x − ∇ V ( x ) · ∂ ξ , and the trapped set at energy E for P is K ( E ) = (cid:8) ( x, ξ ) ∈ p − ( E ); t exp( tH p )( x, ξ ) is bounded (cid:9) . Recall that K ( E ) is compact and stable by the Hamiltonian flow for E > N EXAMPLE OF RESONANCE INSTABILITY 5 V top π x ( γ ) π x ( γ ) supp V ref supp W { V ref ≥ E } π x ( γ ) ( a, Figure 3.
The potentials V = V top + V ref and W .In dimension n = 2, we consider the potential(2.1) V ( x ) = V top ( x ) + V ref ( x ) , as in Figure 3 and described below. On one hand, the potential V top is of the form V top ( x ) = V ( x ) V ( x ) where the functions V • ∈ C ∞ ( R ) are single barriers (see Figure 4) with V ( x ) = E − λ x + O ( x ) and V ( x ) = 1 − λ E x + O ( x ) , near 0 and 0 < λ < λ . In particular, V top is an anisotropic bump, V top ( x ) = E − λ x − λ x + O ( x ) , near 0 and (0 ,
0) is a hyperbolic fixed point for H p . The stable/unstable manifold theoremensures the existence of the incoming/outgoing Lagrangian manifolds Λ ± characterized byΛ ± = (cid:8) ( x, ξ ) ∈ T ∗ R ; exp( tH p )( x, ξ ) → (0 ,
0) as t → ∓∞ (cid:9) . They are stable by the Hamiltonian flow and included in p − ( E ). Eventually, there exist twosmooth functions ϕ ± , defined in a vicinity of 0, satisfying(2.2) ϕ ± ( x ) = ± X j =1 λ j x j + O ( x ) , and such that Λ ± = { ( x, ξ ); ξ = ∇ ϕ ± ( x ) } near (0 , V ( x ) x ∈ R x ∈ R V ( x ) E Figure 4.
The potentials V and V . J.-F. BONY, S. FUJII´E, T. RAMOND, AND M. ZERZERI supp V add O supp V top π x ( γ ) π x ( γ ) π x ( γ )supp W supp V abs ( a, Figure 5.
A realization with a potential V top and an obstacle O .On the other hand, the reflecting potential V ref is non-trapping and localized near ( a, ∈ R with a large enough. A dynamical result (Lemma B.1 of [4]) ensures that no Hamiltoniantrajectory of energy E can start from the support of V ref , touch the support of V top and thencome back to the support of V ref . Thus, a trapped trajectory of energy E is either { (0 , } or a Hamiltonian trajectory starting asymptotically from the origin, touching the support of V ref and coming back to the origin; these latter trajectories are called homoclinic. In otherwords, K ( E ) satisfies K ( E ) = Λ − ∩ Λ + , and H = Λ − ∩ Λ + \ { (0 , } denotes the set of homoclinic trajectories.Giving to V ref the form of a “croissant” barrier, we can make sure that H consists of afinite number of trajectories { γ , . . . , γ K } on which Λ − and Λ + intersect transversally. In thesequel, we will need at least two homoclinic trajectories, that is K ≥
2. Such a geometricconfiguration can also be realized replacing the potential barrier V ref by an obstacle O havingessentially the form of { V ref ( x ) > E } , the operator being P = − h ∆ R \O + V top ( x ) withDirichlet boundary condition (see Figure 5). In that case, one can easily realize a situationwhere K = 3 whereas it seems complicated to have K = 2 (see Example 4.14 of [4]).As explained in the next section (see also Example 4.24 of [4]), the resonance instability isgoverned by the function(2.3) µ ( τ, h ) = Γ (cid:16) λ + λ λ − i τλ (cid:17) e − πτ λ K X k =1 e iA k /h B k e iT k τ , for τ ∈ C , where A k , B k , T k are dynamical quantities related to the curve γ k = ( x k , ξ k ). Werecall quickly how these quantities are defined and send back the reader to [4] for the proofof convergence of the various objects. First, A k = Z γ k ξ · dx, is the action along γ k . Let ν k denote the Maslov index of Λ + along γ k . The function x k ( t )has the following asymptotics x k ( t ) = g k ± e ± λ t + o (cid:0) e ± λ t (cid:1) , N EXAMPLE OF RESONANCE INSTABILITY 7 as t → ∓∞ for some vector g k ± ∈ R . As a matter of fact, g k ± is collinear to the first vectorof the canonical basis (1 ,
0) and do not vanish. Eventually, if γ k ( t, y ) = ( x k ( t, y ) , ξ k ( t, y )) : R × R −→ T ∗ R is a smooth parametrization of Λ + by Hamiltonian curves such that γ k ( t,
0) = γ k ( t ), the limits M + k = lim s →−∞ s(cid:12)(cid:12)(cid:12) det ∂x k ( t, y ) ∂ ( t, y ) | t = s, y =0 (cid:12)(cid:12)(cid:12) e − s λ λ , M − k = lim s → + ∞ s(cid:12)(cid:12)(cid:12) det ∂x k ( t, y ) ∂ ( t, y ) | t = s, y =0 (cid:12)(cid:12)(cid:12) e − s λ − λ , exist and belong to ]0 , + ∞ [. With these notations,(2.4) B k = r λ π M + k M − k e − π ( ν k + ) i (cid:12)(cid:12) g k − (cid:12)(cid:12)(cid:0) iλ | g k + || g k − | (cid:1) − λ λ λ ,T k = ln( λ | g k + || g k − | ) λ . Note that B k ∈ C \ { } and T k ∈ R .The idea is to find a geometric situation and a set H ⊂ ]0 ,
1] with 0 ∈ H such that(2.5) µ ( τ, h ) = 0 , for all τ ∈ C and h ∈ H . For simplicity, we take in the sequel K = 3 as in Figure 3 or 5 andassume that the trajectories γ and γ are symmetric. In particular, A = A , B = B and T = T . We consider two situations:Case (I): A = A (say A > A ), 2 B = B e iν , ν ∈ R , and T = T . Using (2.3) and thesymmetry of γ and γ , these relations imply that (2.5) holds true with H = n A − A (2 j + 1) π + ν ; j ∈ N o . The required relations can be realized since T is only given by the potential V on the line R × { } if ∂ x V ( x ,
0) = 0 for all x ∈ R , whereas B is given by ∂ x V on R × { } . If V ref is replaced by an obstacle O , one may need an additional potential V add in order to satisfythese relations (see Figure 5).Case (II): A = A , 2 B = − B and T = T . In this setting, (2.5) holds true with H =]0 , V ref on R × { } with ∂ x V = 0 on R × { } in order to have A = A and T = T . Then, playingon the Maslov index and on ∂ x V ref on R × { } , one can obtain 2 B = − B .Adding an absorbing potential − ih | ln h | V abs , with V abs ≥
0, it is possible to artificiallyremove a homoclinic trajectory and thus to work with only K = 2 trajectories (see Remark2.1 ii ) and Example 4.14 of [4]). The resulting operator will be non self-adjoint (dissipative)but the conclusions of Theorem 1.1 will still hold.For the perturbation W , we take any non-negative C ∞ ( R ; R ) function supported awayfrom the support of V and non-zero on the base space projection of only one homoclinictrajectory. In the sequel, we will assume that this trajectory is γ as in Figures 3 and 5.We assume that W = 0 near the support of V only to simplify the discussion. The sameway, W ≥ W non-zero on π x ( γ ) can be weakened to R R W ( x ( t )) dt = 0. Finally, J.-F. BONY, S. FUJII´E, T. RAMOND, AND M. ZERZERI W = c W + c W + c W , with W j non-zero only on γ j , may be suitable for Theorem 1.1generically with respect to c j ∈ R .3. Proof of the spectral instability
We consider the operators constructed in the previous section. In particular, we work indimension n = 2, the trapped set of energy E > K = 3 homoclinic trajectories and(2.5) holds true for h ∈ H . Following Chapter 4 of [4], the resonances of P closest to thereal axis are given by the 3 × Q whose entries are(3.1) Q k,ℓ ( z, h ) = e iA k /h Γ (cid:0) S ( z, h ) /λ (cid:1)r λ π M + k M − k e − π ( ν k + ) i (cid:12)(cid:12) g ℓ − (cid:12)(cid:12)(cid:0) iλ | g k + || g ℓ − | (cid:1) − S ( z,h ) /λ , with rescaled spectral parameter(3.2) S ( z, h ) = λ + λ − i z − E h . The same way, the entries of the corresponding matrix for e P = P + h δ W are(3.3) e Q k,ℓ ( z, h ) = ( e − iwh δ Q k,ℓ ( z, h ) if k = 1 , Q k,ℓ ( z, h ) if k = 1 , with the notation w = R R W ( x ( t )) dt = 0. Lemma 3.1.
The matrices Q and e Q are of rank one with non-zero entries. Moreover, Q ( z, h ) = 0 for all z ∈ C and h ∈ H . Proof.
Since M ±• = 0 and g •± = 0, the entries of Q and e Q are always non-zero. From (3.1), theentries of Q can be written Q k,ℓ = α k β ℓ for some α k , β k ∈ C \ { } . In particular, Q = α ( β, · )with α, β ∈ C \ { } and Q is of rank one (the same thing for e Q ). Thus, 0 is an eigenvalue of Q of multiplicity at least 2 and the last eigenvalue is given by its trace, that istr( Q ) = Γ (cid:0) S ( z, h ) /λ (cid:1) X k =1 e iA k /h r λ π M + k M − k e − π ( ν k + ) i (cid:12)(cid:12) g k − (cid:12)(cid:12)(cid:0) iλ | g k + || g k − | (cid:1) − λ λ λ + i z − E λ h = µ (cid:16) z − E h , h (cid:17) . For h ∈ H , all the eigenvalues of Q are zero from (2.5) and Q is nilpotent. Since Q j = α ( β, α ) j − ( β, · ), Q is nilpotent iff ( β, α ) = 0 iff Q = 0. (cid:3) Let W be the 3 × W = diag( − iw, , WQ ( z, h ) are(3.4) − iw Q , ( z, h ) , , . In the present setting, the quantization rule for e P takes the following form: we say that z isa pseudo-resonance of e P when(3.5) 1 ∈ sp (cid:0) h S ( z,h ) /λ − / δ WQ ( z, h ) (cid:1) . The set of pseudo-resonances is denoted by Res ( e P ). Since (3.5) is similar to Definition 4.2 of[4], we can adapt Proposition 4.3 and Lemma 11.3 of [4] in our case and obtain the followingasymptotic of the pseudo-resonances. N EXAMPLE OF RESONANCE INSTABILITY 9
Lemma 3.2.
Let < δ < / , C, β > and ε ( h ) be a function which goes to as h → .Then, uniformly for τ ∈ [ − C, C ] , the pseudo-resonances z of e P in (3.6) E + [ − Ch, Ch ] + i h − (cid:16) λ δλ (cid:17) h − C h | ln h | , h i , with Re z ∈ E + τ h + hε ( h )[ − , satisfy z = z q ( τ ) + o ( h | ln h | − ) with (3.7) z q ( τ ) = E − A λ | ln h | + 2 qπλ h | ln h | − ih (cid:16) λ δλ (cid:17) + i ln( e µ ( τ )) λ h | ln h | , and e µ ( τ ) = w Γ (cid:16) − δ − i τλ (cid:17)r λ π M +1 M − e − π ( ν + ) i (cid:12)(cid:12) g − (cid:12)(cid:12)(cid:0) iλ | g || g − | (cid:1) − + δ + i τλ , for some q ∈ Z . On the other hand, for each τ ∈ [ − C, C ] and q ∈ Z such that z q ( τ ) belongsto (3.6) with a real part lying in E + τ h + hε ( h )[ − , , there exists a pseudo-resonance z satisfying z = z q ( τ ) + o ( h | ln h | − ) uniformly with respect to q, τ . Moreover, there exists M > such that, for all z ∈ (3.6) , we have (3.8) dist (cid:0) z, Res ( e P ) (cid:1) > β h | ln h | = ⇒ (cid:13)(cid:13)(cid:0) − h S/λ − / δ WQ (cid:1) − (cid:13)(cid:13) ≤ M. In the lemma, we have used that the eigenvalues of WQ are explicitly given by (3.4) andthat two of them are zero. On the contrary, note that e µ ( τ ) is a smooth function whichdoes not vanish and that there are a lot of pseudo-resonances in (3.6). The assumption0 < δ < / Lemma 3.3.
For all < δ < / , ν = λ / λ / and C, β > , the following propertiesare satisfied for h ∈ H small enough. i ) For all z ∈ E + [ − Ch, Ch ] + i [ − νh, h ] , we have (3.9) (cid:13)(cid:13)(cid:0) − h S/λ − / Q (cid:1) − (cid:13)(cid:13) . max (cid:0) , h λ λ + Im zλ h (cid:1) .ii ) For all z ∈ (3.6) with dist( z, Res ( e P )) > βh | ln h | − , we have (3.10) (cid:13)(cid:13)(cid:0) − h S/λ − / e Q (cid:1) − (cid:13)(cid:13) . h − δ . The particular value of ν in Lemma 3.3 has no particular meaning. We only need ν >D = λ / ν < λ / λ / Proof.
Since Q = 0 by Lemma 3.1, we get(3.11) (cid:0) − h S/λ − / Q (cid:1) − = 1 + h S/λ − / Q . Using that | h S/λ − / | = h λ λ + Im zλ h and that Q ( z, h ) is uniformly bounded for z ∈ E +[ − Ch, Ch ] + i [ − νh, h ], this identity yields (3.9). On the other hand, (3.3), e − iwh δ = 1 − iwh δ + O ( h δ ) and Q = 0 give1 − h S/λ − / e Q = 1 − h S/λ − / Q − h S/λ − / δ WQ + O ( h λ λ + Im zλ h +2 δ ) Q = (cid:16) − h S/λ − / δ WQ + O ( h λ λ + Im zλ h +2 δ ) Q (cid:17)(cid:0) − h S/λ − / Q (cid:1) = (cid:16) O ( h λ λ + Im zλ h +2 δ ) (cid:0) − h S/λ − / δ WQ (cid:1) − (cid:17)(cid:0) − h S/λ − / δ WQ (cid:1)(cid:0) − h S/λ − / Q (cid:1) . (3.12)We have | h S/λ − / | = h λ λ + Im zλ h ≤ h − δ for z ∈ (3.6). Combining these estimates with (3.8),(3.9) and (3.11), (3.12) implies (cid:0) − h S/λ − / e Q (cid:1) − = (cid:0) − h S/λ − / Q (cid:1) − (cid:0) − h S/λ − / δ WQ (cid:1) − (cid:0) O ( h δ ) (cid:1) − = (cid:0) h S/λ − / Q (cid:1)(cid:0) − h S/λ − / δ WQ (cid:1) − + O (1) , (3.13)if dist( z, Res ( e P )) > βh | ln h | − . Then (3.10) follows. (cid:3) The next result provides a resonance free region for P and the asymptotic of the resonancesclosest to the real axis for e P = P + h δ W . Combined with Lemma 3.2, it implies directlyTheorem 1.1 with D = λ / λ = 1. Lemma 3.4.
There exists α > such that, for all δ > small enough and C > , thefollowing properties hold for h ∈ H small enough. i ) P has no resonance in E + [ − Ch, Ch ] + i h − (cid:16) λ α (cid:17) h, h i .ii ) In the domain (3.6) , we have dist (cid:0)
Res( e P ) , Res ( e P ) (cid:1) = o (cid:16) h | ln h | (cid:17) . As in Definition 4.4 of [4], the notation dist(
A, B ) ≤ ε in C means that ∀ a ∈ A ∩ C, ∃ b ∈ B, | a − b | ≤ ε, and ∀ b ∈ B ∩ C, ∃ a ∈ A, | a − b | ≤ ε. The proof of Lemma 3.4 gives a polynomial estimate of the resolvents in the correspondingdomains (at distance larger than h | ln h | − from the pseudo-resonances of e P ). Proof.
The first point of the lemma has already been obtained in Lemma 12.1 of [4]. In orderto show the second point, we follow the strategy of Chapters 11 and 12 of [4] and summarizedin the introduction of [4]. Then, we first prove that e P has no resonance and we show apolynomial estimate of its resolvent away from the pseudo-resonances. Lemma 3.5.
For δ > small enough, C, β > and h ∈ H small enough, e P has no resonancein the domain (3.14) E + [ − Ch, Ch ] + i h − (cid:16) λ δλ (cid:17) h − C h | ln h | , h i \ (cid:16) Res ( e P ) + B (cid:16) , β h | ln h | (cid:17)(cid:17) , N EXAMPLE OF RESONANCE INSTABILITY 11 ρ k + γ k supp Wρ k − Figure 6.
The geometric setting in the proof of Lemma 3.5. and there exists
M > such that the distorted operator e P θ of angle θ = h | ln h | satisfies (cid:13)(cid:13) ( e P θ − z ) − (cid:13)(cid:13) . h − M , uniformly for h ∈ H small enough and z ∈ (3.14) . Proof of Lemma 3.5.
This result is just an adaptation of Proposition 11.4 of [4]. We onlygive the changes which have to be made in the present setting, sending back the reader toSection 11.2 of [4] for the technical details. From the general arguments of Chapter 8 of [4],it is enough to show that any u = u ( h ) ∈ L ( R ) and z = z ( h ) ∈ (3.14) with(3.15) ( ( e P θ − z ) u = O ( h ∞ ) , k u k L ( R ) = 1 , vanishes microlocally near each point of K ( E ). For k = 1 , ,
3, let u k ± be microlocal restric-tions of u near ρ k ± , where ρ k − (resp. ρ k + ) is a point on γ k just “before” (resp. “after”) (0 , u k − ∈ I (Λ ,k + , h − N ) and u k + ∈ I (Λ , h − N ) , associated to the Lagrangian manifold Λ + just after (0 ,
0) (denoted Λ ) and after a turnalong γ k (denoted Λ ,k + ) for some N ∈ R .After an appropriate renormalization (see [4, (11.25)]), the symbols a k − ( x, h ) ∈ S ( h − N ) of u k − satisfy the relation(3.16) a k − ( x, h ) = h S ( z,h ) /λ − / X ℓ =1 P k,ℓ ( x, h ) a ℓ − ( x ℓ − , h ) + S ( h − N + ζ − δ ) , near x k − = π x ( ρ k − ). In this expression, the symbols P k,ℓ ∈ S (1) (resp. the constant ζ >
0) areindependent of u (resp. δ, u ) and P k,ℓ ( x k − , h ) = e Q k,ℓ ( z, h ). Compared with [4, (11.27)], Q isreplaced by e Q in P k,ℓ ( x k − , h ). Indeed, no change has to be made for the propagation throughthe fixed point (0 ,
0) since W is supported away from V top (see [4, (11.29)]), but the usualtransport equation near γ k ∇ ϕ + · ∇ a k − + (∆ ϕ + − iσ ) a k − = O ( h − N +1 ) , with σ = ( z − E ) /h is replaced by2 ∇ ϕ + · ∇ a k − + (∆ ϕ + − iσ + ih δ W ) a k − = O ( h − N +1 ) , giving on the curve γ k ∂ t a k − ( x k ( t )) + (∆ ϕ + − iσ + ih δ W ) a k − ( x k ( t )) = O ( h − N +1 ) , and leading to the additional factor e − ih δ R W ( x k ( t )) dt in the quantization matrix e Q (see [4,(11.31)]). On the other hand, the remainder term O ( h − N + ζ − δ ) in (3.16) comes from thefact that | h S/λ − / | . h − δ uniformly for z ∈ (3.14) (see Chapter 12.2 of [4] for a similarargument).Applying (3.16) with x = x k − , we get (cid:0) − h S ( z,h ) /λ − / e Q ( z, h ) (cid:1) a − ( x − , h ) = O ( h − N + ζ − δ ) , where a − ( x − , h ) is a shortcut for the 3-vector with coefficients a k − ( x k − , h ). From (3.10), ityields | a − ( x − , h ) | . h − N + ζ − δ , uniformly for z ∈ (3.14). Using again (3.16), we deduce a k − ∈ S ( h − N + ζ − δ ) ⊂ S ( h − N + ζ/ ) for δ > a k − ∈ S ( h − N ), we have proved a k − ∈ S ( h − N + ζ/ ). Bya bootstrap argument (see the end of Chapter 9 of [4]), we obtain u = O ( h ∞ ) microlocallynear K ( E ) and the lemma follows. (cid:3) To finish the proof of Lemma 3.4, it remains to show that e P has a resonance near eachpseudo-resonance. That is Lemma 3.6.
For δ > small enough, C, β > and h ∈ H small enough, the operator e P has at least one resonance in B ( z, βh | ln h | − ) for any pseudo-resonance z ∈ (3.6) . Proof of Lemma 3.6.
This result is equivalent to Proposition 11.6 of [4] in the present setting,and we only explain how to adapt its proof. If Lemma 3.6 did not hold, there would exist asequence z = z ( h ) ∈ (3.6) of pseudo-resonances where h ∈ H goes to 0 such that(3.17) e P has no resonance in D = B (cid:16) z, β h | ln h | (cid:17) . We now construct a “test function”. Let e v be a WKB solution near x − of ( ( e P − e z ) e v = 0 near x − , e v ( x ) = e iϕ , ( x ) /h on | x | = | x − | near x − , for e z ∈ ∂ D , where ϕ , is a generating function of Λ , . Note that e P = P near x − andthat e v can be chosen holomorphic with respect to e z near D . After multiplication by a renor-malization factor as in [4, (11.44)], this function is denoted b v . Consider cut-off functions χ, ψ ∈ C ∞ ( T ∗ R ) such that χ = 1 near ρ − and ψ = 1 near the part of the curve supp( ∇ χ ) ∩ γ before ρ − . Then, we take as “test function” v = Op( ψ ) (cid:2) e P ,
Op( χ ) (cid:3)b v. Let u ∈ L ( R ) be the solution of(3.18) ( e P θ − e z ) u = v, N EXAMPLE OF RESONANCE INSTABILITY 13 for e z ∈ ∂ D . From Lemmas 3.2 and 3.5, u is well-defined and polynomially bounded. Let u k − be a microlocal restriction of u near ρ k − as before. Working as in Lemma 11.10 of [4], one canshow that u k − ∈ I (Λ ,k + , h − δ ) with renormalized symbol a k − . Moreover, as in (3.16), we get(3.19) a k − ( x, h ) = h S ( z,h ) /λ − / X ℓ =1 P k,ℓ ( x, h ) a ℓ − ( x ℓ − , h ) + e a k ( x, h ) + S ( h ζ − δ ) , near x k − , where e a k denotes the symbol of e v near x k − . In particular, e a k ( x, h ) = 0 for k = 1 and e a ( x − , h ) = 1. This relation is obtained using the proofs of (3.16) and Lemma 11.8 of [4].To obtain a contradiction with (3.17), we consider(3.20) I = 12 iπ Z ∂ D u ( e z ) d e z. From the properties of u k − and | ∂ D| = 2 πβh | ln h | − , we have I ∈ I (Λ ,k + , h − δ | ln h | − )microlocally near ρ k − , where its renormalized symbol b k ( x, h ) satisfies(3.21) b k ( x, h ) = 12 iπ Z ∂ D a k − ( x, h ) d e z. Applying (3.19) with x = x k − leads to (cid:0) − h S ( z,h ) /λ − / e Q ( z, h ) (cid:1) a − ( x − , h ) = e a ( x − , h ) + O ( h ζ − δ ) , where c ( x − , h ) is a generic shortcut for the 3-vector with coefficients c k ( x k − , h ). It implies a − ( x − , h ) = (cid:0) − h S/λ − / e Q (cid:1) − e a ( x − , h ) + O ( h ζ − δ )= (cid:0) h S/λ − / Q (cid:1)(cid:0) − h S/λ − / δ WQ (cid:1) − e a ( x − , h ) + O (1) + O ( h ζ − δ ) . from (3.10) and (3.13). We deduce W a − ( x − , h ) = W (cid:0) − h S/λ − / δ WQ (cid:1) − e a ( x − , h ) − h − δ e a ( x − , h )+ h − δ (cid:0) − h S/λ − / δ WQ (cid:1) − e a ( x − , h ) + O (1) + O ( h ζ − δ ) . Inserting this expression in (3.21) and using (3.8) yield W b ( x − , h ) = h − δ iπ Z ∂ D (cid:0) − h S/λ − / δ WQ (cid:1) − e a ( x − , h ) d e z + O (cid:16) h | ln h | (cid:17) + O (cid:16) h ζ − δ | ln h | (cid:17) . Note that e a ( x − , h ) = t (1 , ,
0) is an explicit eigenvector of WQ associated to its non-zeroeigenvalue − iw Q , ( z, h ) (see (3.4)). Thus, computing the integral as in [4, (11.67)], we get W b ( x − , h ) = iλ h − δ | ln h | e a ( x − , h ) + o (cid:16) h − δ | ln h | (cid:17) + O (cid:16) h | ln h | (cid:17) + O (cid:16) h ζ − δ | ln h | (cid:17) . Taking δ > W b ∈ S ( h − δ | ln h | − ), the previous asymptoticshows that W b = 0 so that I = 0. On the other hand, since e P has no resonance in D (see (3.17)), the function u defined by (3.18) is holomorphic in D and (3.20) gives I = 0.Eventually, we get a contradiction and the lemma follows. (cid:3) The second point of Lemma 3.4 is a direct consequence of Lemmas 3.5 and 3.6. (cid:3)
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Jean-Franc¸ois Bony, IMB, CNRS (UMR 5251), Universit´e de Bordeaux, 33405 Talence, France
E-mail address : [email protected] Setsuro Fujii´e, Department of Mathematical Sciences, Ritsumeikan University, 1-1-1 Noji-Higashi, Kusatsu, 525-8577 Japan
E-mail address : [email protected] Thierry Ramond, Universit´e Paris-Saclay, CNRS, Laboratoire de math´ematiques d’Orsay,91405, Orsay, France
E-mail address : [email protected] Maher Zerzeri, Universit´e Sorbonne Paris-Nord, LAGA, CNRS (UMR 7539), 93430 Villeta-neuse, France
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