Widths of resonances above an energy-level crossing
WWIDTHS OF RESONANCES ABOVE AN ENERGY-LEVELCROSSING
S. FUJII´E , A. MARTINEZ AND T. WATANABE Abstract.
We study the existence and location of the resonances of a2 × Keywords:
Resonances; Born-Oppenheimer approximation; eigenvalue cross-ing.
Subject classifications: Department of Mathematical Sciences, RitsumeikanUniversity, 1-1-1 Noji-Higashi, Kusatsu, 525-8577, Japan,[email protected] Universit`a di Bologna, Dipartimento di Matematica, Piazza diPorta San Donato, 40127 Bologna, Italy, [email protected] Department of Mathematical Sciences, RitsumeikanUniversity, 1-1-1 Noji-Higashi, Kusatsu, 525-8577, Japan,[email protected] a r X i v : . [ m a t h - ph ] A p r S. FUJII´E , A. MARTINEZ AND T. WATANABE Introduction
In this paper we continue our study of the resonances for a 2 × h actually represents the square root of thequotient between the electronic and nuclear masses).The situation we investigate is that of two electronic levels that cross at somereal point, corresponding to an energy (say, 0) that is inside the continuousspectrum of the Hamiltonian (see Figure 1). Such a phenomenon occurswhen some energy levels of the molecule cross each other, and this may hap-pen even for diatomic molecules (see, e.g., [Be, DiVi, Le, LeSu] and referencestherein), in which case, due to the rotational invariance of the system, thedimension of the space variable x can be reduced to 1 (so that x actuallyrepresents the distance between the two atoms): see [KMSW, MaSo]. Then,the eigenprojectors can generically be assumed to be smooth, and the re-sulting matrix-potential diagonalizes smoothly, leading to the model we arestudying.Let us also observe that the case of avoided-crossing with a gap of ordersmaller than h also enters our model (in this case, the gap of the avoided-crossing may be included in the coefficient r of the interaction W : seeAssumption (A5)).In [FMW1], we have considered energies very close to 0 (mainly, of size O ( h )), and we have proved the existence of resonances there, and given asharp estimate on their widths in the case of an elliptic interaction at thecrossing point.Then, this result has been extended in [FMW2] to the case of non elliptic in-teractions, that is, more precisely, to the physical case where the interactionconsists of a (non zero) vector field.In both cases, the main technique consists in constructing global WKB solu-tions for the system. This is made possible thanks to appropriate estimateson the fundamental solutions of the scalar Schr¨odinger operators involvedin the problem. (Let us recall that the usual WKB constructions made forscalar operators cannot be extended to systems in general.)Here, we consider the same situation, but this time we investigare the reso-nances E = E ( h ) that have a real part close to some E > O ( h ). Of course, we still use the WKB solutions constructed in[FMW1], but since the region where they oscillate overlap, the behaviours ofthe fundamental solutions are not as good as in the cases of [FMW1, FMW2].However, they are sufficient to prove the existence of resonances with a roughestimate of their location.In order to compute in a precise way their asymptotics, we need to imple-ment microlocal techniques in our methods. Namely, we need to specify themicrolocal behaviour of the resonant states near the two crossing points (in IDTHS OF RESONANCES ABOVE AN ENERGY-LEVEL CROSSING 3 phase-space) of the two characteristic surfaces of the problem (see Figure2). To do so, around each of these points we construct two microlocal basesof solutions with specific properties concerning their microsupports, and weexpress any resonant state in these two basis. After that, the connectionformulas between the bases permit us to obtain a very precise asymptoticexpansion of the resonant states on the outgoing branch of the characteristicset, from which sharp estimates on the resonances (both their real parts andtheir widths) can be derived.The microlocal constructions are made in a spirit similar to that of [ABA](that is, mainly by reducing the operator to a normal form), but with aparticular attention on the choice of the Fourier integral operators used inthe reduction.Let us observe that our final result gives resonance widths of order h , thatshould be compared with the order h found in [FMW1]. This difference canbe explained by the different geometries of the characteristic sets. Indeed, inboth cases they are constituted by two curves, but in the current situationthese curves cross transversally, while in [FMW1] they are tangent to eachother. In some sense, it becomes natural to think that in [FMW1] theparticle escapes more easily than here, which makes its life-time shorter,and thus its resonance width larger.Let us also mention the work [As] where a similar problem is considered, butfor negative energies (in which case the resonance widths are exponentiallysmall).The content of the paper is as follows: In section 2, we state our maintheorem. In section 3, we construct outgoing solutions to our system usingthe method established in [FMW1] and [FMW2]. In section 4, we computethe wronskian between these outgoing solutions to show the existence anda rough estimate of the location of resonances. In section 5, we study thestructure of the space of microlocal solutions near crossing points. In section6, we apply this result to compute the asymptotic behaviour of the resonantstate globally in the phase space. Finally in section 7, we complete the proofof our main theorem. Acknowledgements
This research was supported by the University ofBologna during the visit of the first author in 2018 as well as the JSPSgrant-in-aid for scientific research. We also thank Y.Tsutsumi for his hospi-tality in Kyoto University where part of this work was done and M. Assalfor his valuable remarks after reading the text carefully.
S. FUJII´E , A. MARTINEZ AND T. WATANABE Figure 1: The two potentialsFigure 2: Phase-space picture2.
Assumptions and results
We consider a 2 × P u = Eu, P = (cid:18) P hWhW ∗ P (cid:19) , IDTHS OF RESONANCES ABOVE AN ENERGY-LEVEL CROSSING 5 where D x stands for − i ddx , P j = h D x + V j ( x ) ( j = 1 , W = W ( x, hD x ) isa semiclassical differential operator, and W ∗ is the formal adjoint of W . Westudy the asymptotic distribution of resonances in the semiclassical limit h → + in a neighborhood of a fixed real energy E .We suppose the following conditions on the potentials V ( x ) , V ( x ) (see Fig-ure 1), the positive number E and the interaction W ( x, hD x ): Assumption (A1) V ( x ), V ( x ) are real-valued analytic functions on R ,and extend to holomorphic functions in the complex domain, S = { x ∈ C ; | Im x | < δ (cid:104) Re x (cid:105)} where δ > (cid:104) t (cid:105) := (1 + | t | ) / . Assumption (A2)
For j = 1 , V j admits limits as Re x → ±∞ in S , andthey satisfy, lim Re x →−∞ x ∈S V ( x ) > E ; lim Re x →−∞ x ∈S V ( x ) > E ;lim Re x → + ∞ x ∈S V ( x ) > E ; lim Re x → + ∞ x ∈S V ( x ) < E . Assumption (A3)
There exist three numbers a < b < < c such that V ( a ) = V ( c ) = V ( b ) = E ,V (cid:48) ( a ) < , V (cid:48) ( c ) > , V (cid:48) ( b ) < . and that V > E on ( −∞ , a ) ∪ ( c, + ∞ ) , V < E on ( a, c ) ,V > E on ( −∞ , b ) , V < E on ( b, + ∞ ) . Assumption (A4)
The set { x ∈ R ; V ( x ) = V ( x ) , V ( x ) ≤ E , V ( x ) ≤ E } is reduced to { } , and one has V (0) = V (0) = 0, V (cid:48) (0) > V (cid:48) (0) < j := { ξ + V j ( x ) = E } ( j = 1 ,
2) intersect transversally at (0 , ±√ E ) (see Figure 2). Assumption (A5) W ( x, hD x ) is a first order differential operator, W ( x, hD x ) = r ( x ) + ir ( x ) hD x , where r and r are bounded analytic function on S , are real on the real,and such that W is elliptic at the crossing points (0 , ±√ E ), that is,( r (0) , r (0)) (cid:54) = (0 , . In this situation, in a neighbourhood of the energy E , the spectrum of P is essential only, and the resonances of P can be defined, e.g., as thevalues E ∈ C such that the equation P u = Eu has a non trivial outgoingsolution u , that is, a non identically vanishing solution such that, for some θ > x (cid:55)→ u ( xe iθ ) is in L ( R ) ⊕ L ( R ) (see,e.g., [AgCo, ReSi]). Equivalently, the resonances are the eigenvalues of theoperator P acting on L ( R θ ) ⊕ L ( R θ ), where R θ is a complex distortionof R that coincides with e iθ R for x (cid:29) P ) the set of these resonances. S. FUJII´E , A. MARTINEZ AND T. WATANABE For E ∈ C close enough to E , we define the action, A ( E ) := (cid:90) c ( E ) a ( E ) (cid:112) E − V ( t ) dt, where a ( E ) (respectively c ( E )) is the unique solution of V ( x ) = E close to a (respectively close to c ). In this situation, A ( E ) is an analytic function of E near E and A (cid:48) ( E ) is strictly positive for any real E near E .We also fix δ > C > P lying in the set D h ( δ , C ) given by,(2.2) D h ( δ , C ) := [ E − δ , E + δ ] − i [0 , C h ] . For h > k ∈ Z such that ( k + ) πh belongs to A ([ E − δ , E + 2 δ ]),we set,(2.3) e k ( h ) := A − (cid:18) ( k + 12 ) πh (cid:19) . Then, our main result is,
Theorem 2.1.
Under Assumptions (A1)-(A5), there exists δ > such thatfor any C > , one has, for h > small enough Res ( P ) ∩ D h ( δ , C ) = { E k ( h ); k ∈ Z } ∩ D h ( δ , C ) where the E k ( h ) ’s are complex numbers that satisfy Re E k ( h ) = e k ( h ) + O ( h ) , (2.4) Im E k ( h ) = − C ( e k ( h )) h + O ( h / ) , (2.5) uniformly as h → . Here C ( E ) = πγ A (cid:48) ( E ) (cid:12)(cid:12)(cid:12)(cid:12) r (0) E − sin (cid:18) B ( E ) h + π (cid:19) + r (0) E cos (cid:18) B ( E ) h + π (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) with γ := V (cid:48) (0) − V (cid:48) (0) > and B ( E ) := (cid:90) b ( E ) (cid:112) E − V ( x ) dx + (cid:90) c ( E )0 (cid:112) E − V ( x ) dx, where b ( E ) is the unique root of V ( x ) = E close to b . Remark 2.1.
The physical case corresponds to r = 0 identically (see, e.g. [FMW2] ). In that case, the quantity C ( E ) reduces to, C ( E ) = πγ A (cid:48) ( E ) r (0) E cos (cid:18) B ( E ) h + π (cid:19) . Remark 2.2.
Our theorem is valid above the crossing energy with h -independent positive real part. However, it is interesting to observe the be-haviour of the function C ( E ) for positive E of order h , where the asymp-totics of resonances are studied in [FMW1] and [FMW2] . In this energyregion, the factors E − and E concerning r (0) and r (0) respectively areof order h − and h , and hence their square together with h in (2.5) give IDTHS OF RESONANCES ABOVE AN ENERGY-LEVEL CROSSING 7 h and h , which coincide with the results in [FMW1] and [FMW2] respec-tively. More precisely, substituting λ k ( h ) h / for e k ( h ) , we recover the sameformula in the above papers for the width of resonances. Remark 2.3.
It is important to remark that the function C ( E ) vanishes ona discrete subset of R which is determined by the action B ( E ) . In particularthis set is given by B ( E ) = (cid:18) m − (cid:19) πh, m ∈ Z if r (0) = 0 , and by B ( E ) = (cid:18) m + 14 (cid:19) πh, m ∈ Z if r (0) = 0 . When e k ( h ) takes such an value, the first term of the RHS of(2.5) vanishes, which implies that the width of the corresponding resonanceis smaller than h . This phenomenon can be seen as an effect of the closedtrajectory in the phase space made by Γ and Γ (other than Γ itself ). Background
We fix θ > I L := ( −∞ ,
0] ; I θR := F θ ([0 , + ∞ )) ; F θ ( x ) := x + iθf ( x )where f ∈ C ∞ ([ b, + ∞ ); R + ), f ( x ) = x for x large enough, f ( x ) = 0 for x ∈ [ b, x ∞ ] for some x ∞ > c , and f is chosen in such a way that, for any x ≥ x ∞ , one has,(3.1) Im (cid:90) F θ ( x ) x ∞ (cid:112) E − V ( t ) dt ≥ − Ch, with some positive constant C (see [FMW1]).3.1. Fundamental solutions on I L . For E ∈ D h ( δ , C ) with δ smallenough, let u ± j,L be the solutions to ( P j − E ) u = 0 in ( −∞ ,
0] constructed asin Appendix 2 of [FMW1]. In particular, u − j,L decays exponentially at −∞ ,while u + j,L grows exponentially, and their Wronskian W j,L := W [ u − j,L , u + j,L ]satisfies(3.2) W j,L = − πh (1 + O ( h )) ( h → . In the interior of the interval [Re a ( E ) , Re c ( E )], we also have (see [FMW1,Remark 4.1],(3.3) u − ,L ( x ) = 2 h √ π ( E − V ( x )) − cos (cid:18) A ( E ) + ν ( x ) h − π (cid:19) + O ( h ) , where ν ( x ) := (cid:82) xc ( E ) (cid:112) E − V ( t ) dt . S. FUJII´E , A. MARTINEZ AND T. WATANABE For any k ≥ C kb ( I L ) := { u : I L → C of class C k ; (cid:88) ≤ j ≤ k sup x ∈ I L | u ( j ) ( x ) | < + ∞} , equipped with the norm (cid:107) u (cid:107) C kb ( I L ) := (cid:80) ≤ j ≤ k sup I L | u ( j ) | , and we define afundamental solution K j,L : C b ( I L ) → C b ( I L ) ( j = 1 , , of P j − E on I L by setting, for v ∈ C b ( I L ),(3.4) K j,L [ v ]( x ) := u + j,L ( x ) h W j,L (cid:90) x −∞ u − j,L ( t ) v ( t ) dt + u − j,L ( x ) h W j,L (cid:90) x u + j,L ( t ) v ( t ) dt. Then, K j,L satisfies, ( P j − E ) K j,L = , and, because of the form of the operator W , an integration by parts showsthat we also have, K j,L W, K j,L W ∗ : C b ( I L ) → C b ( I L ) ( j = 1 , . As in [FMW1], in view of the construction of solutions to the system, thekey result is the following proposition:
Proposition 3.1.
One has, (3.5) (cid:107) h K ,L W K ,L W ∗ (cid:107) L ( C b ( I L )) + (cid:107) h K ,L W ∗ K ,L W (cid:107) L ( C b ( I L )) = O ( h ) , (3.6) | hK ,L W ∗ v (0) | + | hK ,L W v (0) | = O (sup I L | v | ) , uniformly with respect to v ∈ C b ( I L ) and h > small enough. Remark 3.2.
Actually, it can also be proved that (cid:107) hK ,L W ∗ (cid:107) L ( C b ( I L )) = O ( h − ) , but for our purpose the better estimate (3.6) on hK ,L W ∗ v (0) isneeded. Proof.
We denote by U ( x, t ) (respectively U ( x, t )) the distributional kernelof h W ,L K ,L W (resp. h W ,L K ,L W ∗ ). An integration by parts showsthat we have,(3.7) U ( x, t ) = (cid:101) U ( x, t ) + hr (0) u − ,L ( x ) u +1 ,L (0) δ t =0 ; U ( x, t ) = (cid:101) U ( x, t ) + hr (0) u − ,L ( x ) u +2 ,L (0) δ t =0 , with, (cid:101) U ( x, t ) := u +1 ,L ( x )( W u − ,L )( t ) { t 0] the functions u ± ,L are oscil-lating, and we have (see, e.g., [FMW1], Section 8), | u ± ,L ( s ) | + | h ( u ± ,L ) (cid:48) ( s ) | = O ( h | s − b ( E ) | − ) . , A. MARTINEZ AND T. WATANABE As a consequence, (cid:90) b ( E ) (cid:101) U (0 , s ) v ( s ) ds = O ( h ) sup I L | v | , and thus,(3.13) (cid:90) −∞ (cid:101) U (0 , s ) v ( s ) ds = O ( h ) sup I L | v | . Inserting into (3.12), we obtain,(3.14) sup I L | A | = O ( h ) sup I L | v | . Concerning A , since | u +2 ,L (0) | = O ( h ) uniformly, we have,(3.15) A ( x ) = O ( h ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) −∞ (cid:101) U ( x, t ) u − ,L ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12) sup I L | v | . We prove, Lemma 3.3. (cid:90) −∞ (cid:101) U ( x, t ) u − ,L ( t ) dt = O ( h ) . Proof. Using that (cid:101) U ( x, t ) = O (1) uniformly on ( −∞ , Re c ( E )], and that,for any δ > 0, there exists α > u − ,L ( t ) = O ( h e − α | t | /h ) in ( −∞ , Re b ( E ) − δ ]; u − ,L ( t ) = O ( h | b ( E ) − t | − e − α | b ( E ) − t | /h ) in [Re b ( E ) − δ, Re b ( E ) − h ]; u − ,L ( t ) = O (1) in ( −∞ , Re c ( E )] , we immediately obtain, (cid:90) Re b ( E ) −∞ (cid:101) U ( x, t ) u − ,L ( t ) dt = O ( h ) . On the other hand, when t ∈ [Re b ( E ) , (cid:101) U ( x, t ) = O ( h / ) and u − ,L ( t ) = O ( h / | t − b ( E ) | − / ). Therefore, (cid:90) b ( E ) (cid:101) U ( x, t ) u − ,L ( t ) dt = O ( h ) , and the result follows. (cid:3) Going back to (3.15), this gives us,(3.16) sup I L | A | = O ( h ) sup I L | v | . Concerning A , we first observe that, by the same estimates as in [FMW1],Section 3 (see [FMW1], proof of Proposition 3.1), we have, (cid:90) Re b ( E ) −∞ (cid:90) Re b ( E ) −∞ (cid:12)(cid:12)(cid:12) (cid:101) U ( x, t ) (cid:101) U ( t, s ) (cid:12)(cid:12)(cid:12) dsdt = O ( h ) IDTHS OF RESONANCES ABOVE AN ENERGY-LEVEL CROSSING 11 uniformly for x ∈ ( −∞ , 0] and h > x ∈ I L (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) Re b ( E ) −∞ (cid:90) Re b ( E ) −∞ (cid:101) U ( x, t ) (cid:101) U ( t, s ) v ( s ) dsdt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O ( h ) sup I L | v | . It remains to study the three quantities, A , ( x ) := (cid:90) b ( E ) dt (cid:90) b ( E ) (cid:101) U ( x, t ) (cid:101) U ( t, s ) v ( s ) ds ; A , ( x ) := (cid:90) Re b ( E ) −∞ dt (cid:90) b ( E ) (cid:101) U ( x, t ) (cid:101) U ( t, s ) v ( s ) ds ; A , ( x ) := (cid:90) b ( E ) dt (cid:90) Re b ( E ) −∞ (cid:101) U ( x, t ) (cid:101) U ( t, s ) v ( s ) ds. For A , ( x ), since t ≤ s on the domain of integration, we have, A , ( x ) = (cid:90) Re b ( E ) −∞ (cid:101) U ( x, t ) u − ,L ( t ) dt (cid:90) b ( E ) ( W u +2 ,L )( s ) v ( s ) ds, and thus, since W u +2 ,L is O ( h / | s − b ( E ) | − / ) on [Re b ( E ) , A , ( x ) = O ( h / ) sup I L | v | (cid:90) Re b ( E ) −∞ (cid:101) U ( x, t ) u − ,L ( t ) dt. Thanks to the exponential decay of u − ,L on ( −∞ , Re b ( E )), we immediatelysee that, for any δ > 0, we have,(3.19) (cid:90) Re b ( E ) − δ −∞ (cid:101) U ( x, t ) u − ,L ( t ) dt = O ( e − α/h ) , with α = α ( δ ) > 0. On the other hand, if δ is sufficiently small, for t ∈ [Re b ( E ) − δ, Re b ( E )], we have, (cid:101) U ( x, t ) = O ( h ); u − ,L ( t ) = O ( h | t − b ( E ) | − e − β (Re b ( E ) − t ) / /h ) , where β > (cid:90) Re b ( E )Re b ( E ) − δ (cid:101) U ( x, t ) u − ,L ( t ) dt = O ( h ) (cid:90) δ − δ t − e − βt / /h dt = O ( h / ) , and we finally obtain,(3.20) sup I L | A , | = O ( h ) sup I L | v | . Concerning A , ( x ), we have s ≤ t on the domain of integration, and thus, A , ( x ) = (cid:90) b ( E ) (cid:101) U ( x, t ) u +2 ,L ( t ) dt (cid:90) Re b ( E ) −∞ ( W u − ,L )( s ) v ( s ) ds. , A. MARTINEZ AND T. WATANABE Since (cid:82) Re b ( E ) −∞ | W u − ,L ( s ) | ds = O ( h ) (this can be seen, e.g., as in [FMW1],Section 3), we deduce,(3.21) A , ( x ) = O ( h sup I L | v | ) (cid:90) b ( E ) (cid:101) U ( x, t ) u +2 ,L ( t ) dt. On the other hand, for t ∈ [Re b ( E ) , (cid:101) U ( x, t ) u +1 ,L ( t ) = O ( h / | t − b ( E ) | − / ) , and thus(3.22) A , = O ( h ) sup I L | v | . Concerning A , ( x ), we write,(3.23) A , ( x ) = A +1 , ( x ) + A − , ( x )with, A ± , ( x ) := (cid:90) b ( E ) (cid:101) U ( x, t ) u ± ,L ( t ) w ± ( t ) dt ; w + ( t ) := (cid:90) t Re b ( E ) ( W u − ,L )( s ) v ( s ) ds ; w − ( t ) := (cid:90) t ( W u +2 ,L )( s ) v ( s ) ds, and we observe that, since | u ± ,L | + | W u ± ,L | = O ( h | E − V | − ) and ( E − V ) − is integrable on [Re b ( E ) , w ± ( t ) = O ( h ) sup I L | v | . In addition, by definition, we also have,(3.25) w (cid:48)± ( t ) = O (sup I L | v | ) . Now, we fix λ ≥ A ± , ( x ) = (cid:90) Re b ( E )+ λh / Re b ( E ) (cid:101) U ( x, t ) u ± ,L ( t ) w ± ( t ) dt + (cid:90) b ( E )+ λh / (cid:101) U ( x, t ) u ± ,L ( t ) w ± ( t ) dt, and thus, using (3.24) and the fact that (cid:101) U ( x, t ) = O ( h / ) and u ± ,L ( t ) = O ( h / | t − b ( E ) | − / ) when t ∈ [Re b ( E ) , Re b ( E ) + λh / ], we obtain(3.27) A ± , ( x ) = (cid:90) b ( E )+ λh / (cid:101) U ( x, t ) u ± ,L ( t ) w ± ( t ) dt + O ( h ) sup I L | v | . We first assume x ≤ Re b ( E ). In this case, one has x ≤ t on the domain ofintegration, and thus,(3.28) A ± , ( x ) = C ( v ) u − ,L ( x ) + O ( h ) sup I L | v | . IDTHS OF RESONANCES ABOVE AN ENERGY-LEVEL CROSSING 13 with, C ( v ) := (cid:90) b ( E )+ λh / W u +1 ,L ( t ) u ± ,L ( t ) w ± ( t ) dt Now, on the interval [Re b ( E ) + λh / , V , we can use the WKB expansion of u +1 ,L ( t ),(3.29) u − ,L ( t ) = 2 h √ π ( E − V ( t )) − sin (cid:16) h − ν ( t ) + π (cid:17) + O ( h ); h ( u − ,L ) (cid:48) ( t ) = 2 h √ π ( E − V ( t )) cos (cid:16) h − ν ( t ) + π (cid:17) + O ( h ) , where we have set,(3.30) ν ( t ) := (cid:90) ta ( E ) (cid:112) E − V ( s ) ds. On the other hand, concerning u − ,L , we have (see, e.g., [FMW1, Section 8],and [Ya]),(3.31) u − ,L ( t ) = 2( ξ (cid:48) ( t )) − ˇAi ( h − ξ ( t )) + O ( h ) , where Ai is the usual Airy functions, ˇAi ( t ) := Ai ( − t ), and where ξ = ξ ( t ; E ) is the analytic continuation to complex values of E of the functiondefined for E real by, ξ ( t ; E ) := (cid:32) (cid:90) tb ( E ) (cid:112) E − V ( s ) ds (cid:33) when t ≥ b ( E ); ξ ( t ; E ) := − (cid:32) (cid:90) b ( E ) t (cid:112) V ( s ) − Eds (cid:33) when t ≤ b ( E ) . By choosing λ sufficiently large, we also see that h − ξ ( t ) becomes arbitrar-ily large when t ∈ [Re b ( E )+ λh / , y ) = 1 √ π y − sin (cid:18) y + π (cid:19) + O ( | y | − − );valid uniformly as | y | → ∞ , | arg y | ≤ π − δ ( δ > u − ,L ( t ) = 2 h / ( ξ (cid:48) ( t )) − √ π ( ξ ( t )) sin (cid:32) ξ ( t ) h + π (cid:33) + O (cid:32) h / | ξ ( t ) | + (cid:33) + O ( h ) . Then, using that | ξ ( E ) | behaves like | t − b ( E ) | on this interval, we obtain,(3.32) C ( v ) = C + ( v ) + C − ( v ) + R ( v )where C ± ( v ) = (cid:90) b ( E )+ λh / h a ± ( t ) e ± iν ( t ) /h ( t − b ( E )) / sin (cid:32) ξ ( t ) h + π (cid:33) w ± ( t ) dt , A. MARTINEZ AND T. WATANABE with a ± ( t ) smooth, and where, R ( v ) = (cid:90) b ( E )+ λh / (cid:32) O ( h + ) | t − b ( E ) | / + O ( h + ) | t − b ( E ) | + + O ( h ) (cid:33) | w ± ( t ) | dt. By the same arguments as before (in particular (3.24)), we obtain,(3.33) R ( v ) = O ( h + h + h ) sup I L | v | = O ( h ) sup I L | v | . On the other hand, setting,(3.34) ν ( t ) := (cid:90) tb ( E ) (cid:112) E − V ( s ) ds, we see that C + ( v ) and C − ( v ) are sums of terms of the type, B + = (cid:90) b ( E )+ λh / h a ( t ) e ± i ( ν ( t )+ ν ( t )) /h ( t − b ( E )) / w ± ( t ) dt, or of the type, B − = (cid:90) b ( E )+ λh / h a ( t ) e ± i ( ν ( t ) − ν ( t )) /h ( t − b ( E )) / w ± ( t ) dt, with a ( t ) smooth, and where the various ± are not related each other.Here we observe that, for t ∈ [Re b ( E ) + λh / , • | Im ν ( t ) | + | Im ν ( t ) | = O ( h ); • Re( ν (cid:48) ( t ) + ν (cid:48) ( t )) = Re( (cid:112) E − V ( t ) + (cid:112) E − V ( t )) ≥ C for someconstant C > • Re( ν (cid:48) ( t ) − ν (cid:48) ( t )) = Re( (cid:112) E − V ( t ) − (cid:112) E − V ( t )) vanishes at t = 0only; • | Re( ν (cid:48)(cid:48) ( t ) − ν (cid:48)(cid:48) ( t )) | ≥ C for some constant C > B + -type terms, we write,(3.35) e ± i ( ν ( t )+ ν ( t )) /h = ± hi ( ν (cid:48) ( t ) + ν (cid:48) ( t )) ddt ( e ± i ( ν ( t )+ ν ( t )) /h ) , and we make an integration by parts. Using the notation ϕ := ± ( ν + ν ),we obtain, B + = O ( h ) sup I L | v | + ih (cid:90) b ( E )+ λh / e iϕ ( t ) /h ddt (cid:32) a ( t ) w ± ( t )( t − b ( E )) ϕ (cid:48) ( t ) (cid:33) dt and thus, using (3.25) and the fact that ddt (cid:18) a ( t )( t − b ( E )) ϕ (cid:48) ( t ) (cid:19) is O ( | t − b ( E ) | − ),(3.36) B + = O ( h ) sup I L | v | . Concerning the B − -type terms, in view of performing a stationary-phaseargument, let χ ∈ C ∞ (Re b ( E ) , 0) be a ( h -independent) cut-off function, IDTHS OF RESONANCES ABOVE AN ENERGY-LEVEL CROSSING 15 such that χ = 1 near 0. We write B − = B − , + B − , , with, B − , := (cid:90) b ( E )+ λh / h (1 − χ ( t )) a ( t ) e ± i ( ν ( t ) − ν ( t )) /h ( t − b ( E )) / w ± ( t ) dt, ; B − , := (cid:90) b ( E )+ λh / h χ ( t ) a ( t ) e ± i ( ν ( t ) − ν ( t )) /h ( t − b ( E )) / w ± ( t ) dt. Exactly as for B + , we see,(3.37) B − , = O ( h ) sup I L | v | . In order to estimate B − , , we need some special version of the stationary-phase theorem (it is probably well-known, but we did not find any referencefor it). Lemma 3.4. Let χ ∈ C ∞ ( R ; [0 , with χ = 1 near 0, and ψ ∈ C ∞ ( R ; R ) admitting 0 as unique stationary point in Supp χ with ψ (cid:48)(cid:48) (0) (cid:54) = 0 . Then,denoting by K the convex hull of Supp χ and by sgn ψ (cid:48)(cid:48) (0) the sign of ψ (cid:48)(cid:48) (0) ,one has, for f ∈ C ( R ) , (3.38) (cid:90) e iψ ( t ) /h χ ( t ) f ( t ) dt = f (0) e i π sgn ψ (cid:48)(cid:48) (0) (cid:115) πh | ψ (cid:48)(cid:48) (0) | + O ( h ) sup K ( | f (cid:48) | + | f (cid:48)(cid:48) | ) , uniformly with respect to h > small enough. Proof. First of all, by a smooth change of variable (depending only on ψ ),we can assume that ψ = ± µt / µ > f ( t ) = f (0) + tg ( t )with, g ( t ) := (cid:90) f (cid:48) ( θt ) dθ, we obtain, (cid:90) e iψ/h χ f dt = f (0) (cid:90) e ± iµt / h χ dt ± hiµ (cid:90) ddt ( e ± iµt / h ) χ gdt. By the standard stationary-phase theorem (see, e.g., [Ma]), we have,(3.39) (cid:90) e ± iµt / h χ ( t ) dt = µ − e ± i π √ πh + O ( h ∞ ) , and thus, by an integration by parts, we obtain,(3.40) (cid:90) e iψ/h χ f dt = f (0)( µ − e ± i π √ πh + O ( h ∞ )) ± ihµ (cid:90) e ± iµt / h ddt ( χ g ) dt, where the O ( h ∞ ) does not depend on f . Then, the result follows from thefact that sup Supp χ ( | g | + | g (cid:48) | ) ≤ sup K ( | f (cid:48) | + | f (cid:48)(cid:48) | ). (cid:3) , A. MARTINEZ AND T. WATANABE Remark 3.5. The formula (3.38) stays valid when the integration is re-stricted to the half line R − or R + just by replacing the first term of the RHSwith its half. To see this, it is enough to check, instead of (3.39), that (cid:90) R ± e ± iµt / h χ ( t ) dt = 12 µ − e ± i π √ πh + O ( h ∞ ) , (see also Lemma 3.8 for more general cases) and that the endpoint term ihµ g (0) arising from the integration by parts is of O ( h ) sup K | f (cid:48) | . Lemma 3.6. Let I ⊂ R be an open interval containing 0, and ψ ∈ C ∞ ( R ; R ) admitting 0 as unique stationary point in ¯ I with ψ (cid:48)(cid:48) (0) (cid:54) = 0 . Then, one has,for u ∈ C ( I ) , (cid:90) e iψ ( t ) /h u ( t ) dt = O ( √ h ) sup( | u | + | u (cid:48) | ) , uniformly with respect to h > small enough. Proof. Setting, f ( t ) := 1 √ πh (cid:90) e − ( t − s ) / h u ( s ) ds, we have, f ( t ) − u ( t ) = 1 √ πh (cid:90) e − ( t − s ) / h ( u ( s ) − u ( t )) ds, and thus, since | u ( s ) − u ( t ) | ≤ | s − t | sup | u (cid:48) | and (cid:82) e − ( s − t ) / h | s − t | ds = 2 h , | f ( t ) − u ( t ) | ≤ (cid:114) hπ sup | u (cid:48) | . As a consequence, fixing χ ∈ C ∞ ( R ; [0 , χ = 1 on I (so that χ u = u ), we have,(3.41) (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) e iψ ( t ) /h u ( t ) dt − (cid:90) e iψ ( t ) /h χ ( t ) f ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ | I | (cid:114) hπ sup | u (cid:48) | . On the other hand, applying Lemma 3.4 to f , we have,(3.42) (cid:90) e iψ ( t ) /h χ ( t ) f ( t ) dt = f (0) (cid:115) πh | ψ (cid:48)(cid:48) (0) | + O ( h ) sup( | f (cid:48) | + | f (cid:48)(cid:48) | ) . Now, we can write, f (cid:48) ( t ) = 1 √ πh (cid:90) e − ( t − s ) / h u (cid:48) ( s ) ds ; f (cid:48)(cid:48) ( t ) = − h − √ πh (cid:90) e − ( t − s ) / h ( t − s ) u (cid:48) ( s ) ds. Therefore, | f (cid:48) ( t ) | ≤ sup | u (cid:48) | , and, | f (cid:48)(cid:48) ( t ) | ≤ √ πh sup | u (cid:48) | . Since also | f (0) | ≤ sup | u | , the result follows form (3.41)-(3.42). (cid:3) IDTHS OF RESONANCES ABOVE AN ENERGY-LEVEL CROSSING 17 Remark 3.7. As in Remark 3.5, this lemma remains true when the inte-gration is restricted to a half line R ± . Applying the previous lemma and remark with ψ = ± ( ν − ν ) , u ( t ) = h a ( t )( t − b ( E )) / w ± ( t ) , we obtain, B − , = O ( h ) sup Supp χ ( | w ± | + | w (cid:48)± | ) , and thus, by (3.24) and the fact that, on Supp χ , w (cid:48)± ( t ) = O ( h ) v ( t ),(3.43) B − , = O ( h ) sup I L | v | . Gathering (3.28), (3.32), (3.33), (3.36), (3.37) and (3.43), and using the factthat u − ,L is uniformly bounded on I L , we obtain,(3.44) sup ( −∞ , Re b ( E )] | A ± , | = O ( h ) sup I L | v | . Now, when x ∈ [Re b ( E ) , A ± , ( x ) must bewritten as, A ± , ( x ) = u +1 ,L ( x ) (cid:90) x Re b ( E ) ( W u − ,L )( t ) u ± ,L ( t ) w ± ( t ) dt + u − ,L ( x ) (cid:90) x ( W u +1 ,L )( t ) u ± ,L ( t ) w ± ( t ) dt, but since u +1 ,L is uniformly bounded on [Re b ( E ) , x remains away from thecritical point of ν − ν (in this case, it suffices to choose χ in such a waythat x / ∈ Supp χ ). When x is closed to 0, there are two changes in the proofof Lemma 3.4. The first one is that an extra-term appears in (3.40), theboundary term ∓ ihµ − e ± iµ ˜ x / h χ (˜ x ) g (˜ x ) (where ˜ x is the value of x afterthe change of variable that transforms ψ into ± µt / O ( h ) sup K | f (cid:48) | . The other change concerns (3.39), since we now haveto estimate (cid:82) ± t ≥± ˜ x e ± iµt / h χ ( t ) dt . We need to prove, Lemma 3.8. For any µ > independent of h , a ∈ R (which may dependon h ) and χ as in Lemma 3.4, one has, (cid:90) ± t ≥± a e ± iµt / h χ ( t ) dt = O ( √ h ) . uniformly with respect to h > small enough. Proof. We treat the case t ≥ a only (the other one being similar), and wecan assume that a ∈ Supp χ (otherwise we already know that the integral , A. MARTINEZ AND T. WATANABE is O ( √ h )). We fix some σ > a ≥ − h σ , we write, (cid:90) t ≥ a e ± iµt / h χ ( t ) dt = (cid:90) a +2 h σ a e ± iµt / h χ ( t ) dt + (cid:90) t ≥ a +2 h σ e ± iµt / h χ ( t ) dt = O ( h σ ) ± hiµ (cid:90) t ≥ a +2 h σ ddt (cid:16) e ± iµt / h (cid:17) χ ( t ) t dt = O ( h σ + h − σ ) ± ihµ (cid:90) t ≥ a +2 h σ e ± iµt / h ddt (cid:18) χ ( t ) t (cid:19) dt = O ( h σ + h − σ ) + O ( h ) (cid:90) + ∞ a +2 h σ dtt = O ( h σ + h − σ ) . If a ≤ − h σ , we write, (cid:90) t ≥ a e ± iµt / h χ ( t ) dt = (cid:90) − h σ a e ± iµt / h χ ( t ) dt + (cid:90) t ≥− h σ e ± iµt / h χ ( t ) dt = (cid:90) − h σ a e ± iµt / h χ ( t ) dt + O ( h σ + h − σ )= ± hiµ (cid:90) − h σ a ddt (cid:16) e ± iµt / h (cid:17) χ ( t ) t dt + O ( h σ + h − σ )= O ( h σ + h − σ ) , where the last estimates comes again from an integration by parts. Taking σ = , the result follows. (cid:3) Hence, the estimate remains the same as in Lemma 3.6, and we finallyobtain,(3.45) sup I L | A , | = O ( h ) sup I L | v | . Then, the required estimate on the norm of h K ,L W K ,L W ∗ follows from(3.8), (3.9), (3.11), (3.14), (3.16), (3.17), (3.20), (3.22) and (3.45). The samearguments also apply to h K ,L W ∗ K ,L W , and this proves (3.5).Concerning (3.6), by using (3.7) we have, hK ,L W ∗ v (0) = O ( h − ) (cid:90) −∞ (cid:101) U (0 , t ) v ( t ) dt + O ( h ) r (0) u − ,L (0) u +2 ,L (0) v (0) , and thus, using the fact that | u ± ,L ( t ) | + | W u ± ,L ( t ) | = O ( h ) | t − b ( E ) | − e − h (cid:82) b ( E ) t √ ( V ( s ) − E ) + ds , IDTHS OF RESONANCES ABOVE AN ENERGY-LEVEL CROSSING 19 we obtain, hK ,L W ∗ v (0) = O ( h − ) (cid:90) −∞ (cid:101) U (0 , t ) v ( t ) dt + O ( h ) sup I L | v | = O ( h − ) (cid:90) −∞ W u − ,L ( t ) v ( t ) dt + O ( h ) sup I L | v | = O (sup I L | v | ) . Similar arguments hold for hK ,L W v (0), and Proposition 3.1 follows. (cid:3) Fundamental solutions on I θR . Exactly as in [FMW1], the sameconstructions hold on I θR , and lead to fundamental solutions K j,R : C b ( I θR ) → C b ( I θR ) ( j = 1 , , of P j − E on I θR . Then, the same arguments as in the previous section alsogive, Proposition 3.9. One has, (3.46) (cid:107) h K ,R W ∗ K ,R W (cid:107) L ( C b ( I θR )) + (cid:107) h K ,R W K ,R W ∗ (cid:107) L ( C b ( I θR )) = O ( h ) , (3.47) | hK ,R W v (0) | + | hK ,R W ∗ v (0) | = O (sup I θR | v | ) , uniformly with respect to v ∈ C b ( I θR ) and h > small enough. Solutions of the system on I L and I θR . Following [FMW1, Section4], we set, M L := h K ,L W K ,L W ∗ ; (cid:102) M L := h K ,L W ∗ K ,L WM R := h K ,R W ∗ K ,R W ; (cid:102) M R := h K ,R W K ,R W ∗ and, thanks to Propositions 3.1 and 3.9, we see that the convergent seriesgiven by,(3.48) w ,L := (cid:32) (cid:80) j ≥ M jL u − ,L − hK ,L W ∗ (cid:80) j ≥ M jL u − ,L (cid:33) ; w ,L := (cid:32) − hK ,L W (cid:80) j ≥ (cid:102) M jL u − ,L (cid:80) j ≥ (cid:102) M jL u − ,L , (cid:33) are solutions to (2.1) on I L , while the convergent series,(3.49) w ,R := (cid:32) (cid:80) j ≥ (cid:102) M jR u − ,R − hK ,R W ∗ (cid:80) j ≥ (cid:102) M jR u − ,R (cid:33) ; w ,R := (cid:32) − hK ,R W (cid:80) j ≥ M jR u − ,R (cid:80) j ≥ M jR u − ,R (cid:33) , A. MARTINEZ AND T. WATANABE (where u − j,R are the solutions to ( P j − E ) u − j,R = 0 constructed in [FMW1,Appendix 2]) are solutions to (2.1) on I θR . In addition, one also has (see[FMW1, Proposition 4.1]),(3.50) w j,L ∈ L ( I L ) ⊕ L ( I L ) ; w j,R ∈ L ( I θR ) ⊕ L ( I θR ) . Existence and location of resonances The four solutions constructed in the previous section permits us to writethe quantization condition that determines the resonances of P in D h ( δ , C )as,(4.1) W ( E ) = 0 , where W ( E ) := W ( w ,L , w ,L , w ,R , w ,R ) stands for the Wronskian of w ,L , w ,L , w ,R and w ,R . Since this Wronskian is constant with respect to x , weplan to compute it at x = 0. We first show, Proposition 4.1. For S = L, R , one has, w ,S (0) = (cid:18) u − ,S (0)0 (cid:19) + O ( h ) ; w (cid:48) ,S (0) = (cid:18) ( u − ,S ) (cid:48) (0)0 (cid:19) + O ( h − ); w ,S (0) = (cid:18) u − ,S (0) (cid:19) + O ( h ) ; w (cid:48) ,S (0) = (cid:18) u − ,S ) (cid:48) (0) (cid:19) + O ( h − ) . Proof. We write the proof for j = 1 and S = L , the ones for j = 2 or S = R being similar. By Proposition 3.1 and (3.48), we have,(4.2) w ,L (0) = (cid:18) u − ,L (0) − hK ,L W ∗ u − ,L (0) (cid:19) + O ( h ) . On the other hand, by (3.7) and (3.2)-(3.4), hK ,L W ∗ u − ,L (0) = O ( h − ) (cid:90) −∞ (cid:101) U (0 , t ) u − ,L ( t ) dt + O ( h ) r (0) u − ,L (0) u +2 ,L (0) u − ,L (0) , and thus, using the fact that | u ± j,L (0) | = O ( h ),(4.3) hK ,L W ∗ u − ,L (0) = O ( h − ) (cid:90) −∞ (cid:101) U (0 , t ) u − ,L ( t ) dt + O ( h ) . Now, we show, Lemma 4.2. (cid:90) −∞ (cid:101) U (0 , t ) u − ,L ( t ) dt = O ( h ) . IDTHS OF RESONANCES ABOVE AN ENERGY-LEVEL CROSSING 21 Proof. Since W u − ,L ( t ) is exponentially small on ( −∞ , Re b ( E ) − δ ] ( δ > t and h , we have,(4.4) (cid:90) −∞ (cid:101) U (0 , t ) u − ,L ( t ) dt = u +2 ,L (0) (cid:90) −∞ W u − ,L ( t ) u − ,L ( t ) dt = O ( h ) (cid:90) b ( E ) − δ W u − ,L ( t ) u − ,L ( t ) dt + O ( e − α/h )with α = α ( δ ) > 0. Then, using the asymptotic behaviour of u − ,L and u − ,L near b ( E ) (see, e.g., [FMW1, Appendix 2]), we have (for some β > (cid:90) Re b ( E )Re b ( E ) − δ W u − ,L ( t ) u − ,L ( t ) dt = O ( h ) (cid:90) Re b ( E )Re b ( E ) − δ e − β | t − b ( E ) | / /h | t − b ( E ) | / dt = O ( h / ) , and, for any fixed λ > (cid:90) Re b ( E )+ λh Re b ( E ) W u − ,L ( t ) u − ,L ( t ) dt = O ( h ) (cid:90) Re b ( E )+ λh Re b ( E ) dt = O ( h / ) . Moreover, if λ has been taken large enough, we also have (see, e.g., [FMW1,Propositions A.5]), (cid:90) b ( E )+ λh W u − ,L ( t ) u − ,L ( t ) dt = (cid:88) σ ∈{± } (cid:90) b ( E )+ λh h f σ ( t )( t − b ( E )) e iν σ ( t ) /h dt + O ( h ) , where f σ is smooth and bounded (together with all its derivatives), andwhere, for σ = ( σ , σ ) ∈ {± } we have used the notation, ν σ := σ ν + σ ν . (Here, ν and ν are the functions defined in (3.30) and (3.34).) In particular,we see that ν σ has no critical point in [Re b ( E ) + λh , − δ ], and thus, writing e iν σ ( t ) /h = hiν (cid:48) σ ( t ) ddt e iν σ ( t ) /h , and integrating by parts, we obtain,(4.7) (cid:90) − δ Re b ( E )+ λh W u − ,L ( t ) u − ,L ( t ) dt = O ( h ) + O ( h ) (cid:90) − δ Re b ( E )+ λh ( t − b ( E )) − dt = O ( h ) . It remains us to study (cid:82) − δ W u − ,L ( t ) u − ,L ( t ) dt . Since the phase functions ν σ corresponding to σ = ± (1 , − 1) have a non-degenerate critical point at 0,after a change of variables the corresponding integrals can be transformedinto, h (cid:90) δ (cid:48) f ( t ) e it / h dt, , A. MARTINEZ AND T. WATANABE with f smooth and bounded together with all its derivatives, and δ (cid:48) > (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) − δ W u − ,L ( t ) u − ,L ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12) = O ( h ) , and by (4.4) Lemma 4.2 follows. (cid:3) Going back to (4.2) and (4.3), the previous lemma implies,(4.8) w ,L (0) = (cid:18) u − ,L (0)0 (cid:19) + O ( h ) . Concerning w (cid:48) ,L (0), we just observe that for j = 1 , 2, the function hD x u ± j,L have a behaviour of the same type as u ± j,L , and the same computations as inthe proof of Proposition 3.1 also give, (cid:107) h D x K ,L W K ,L W ∗ (cid:107) L ( C b ( I L )) = O ( h − ); hD x K ,L W ∗ v (0) = O ( h − ) sup I L | v | . (Observe that, when differentiating K j,L v ( x ), the terms that come out whenthe derivative acts on the x of (cid:82) x or (cid:82) x cancel each other.)For the same reason, we also obtain, hD x K ,L W ∗ u − ,L (0) = O ( h − ) . Thus, in the same way as for w ,L (0) (that is, using (3.48)), we have,(4.9) w (cid:48) ,L (0) = (cid:18) ( u − ,L ) (cid:48) (0)0 (cid:19) + O ( h − ) . Bu using (3.48)-(3.49) and Proposition 3.5, we can repeat the same argu-ments for j = 2 and/or S = R , and Proposition 4.1 follows. (cid:3) Proposition 4.3. For any E ∈ D h ( δ , C ) , one has, W ( E ) = 4 √ π e − i π h − cos A ( E ) h + O ( h − ) , uniformly as h → + . Proof. Using Proposition 4.1, and the fact that, for S = L, R and j = 1 , u j,S (0) = O ( h ) and ( u j,S ) (cid:48) (0) = O ( h − ), we immediately obtain, W ( E ) = W ( u − ,L , u − ,R ) W ( u − ,L , u − ,R ) + O ( h − ) . On the other hand, we know from standard WKB constructions (see also[FMW1, Appendix]) that we have,(4.10) W ( u − ,L , u − ,R ) = − π h − cos A ( E ) h + O ( h ); W ( u − ,L , u − ,R ) = i √ π h − e i π + O ( h ) , and the result follows. (cid:3) IDTHS OF RESONANCES ABOVE AN ENERGY-LEVEL CROSSING 23 Now, we are able to establish the existence of resonances, together with apreliminary (but fundamental) result on their location. With the definitionof e k ( h ) given in (2.3), we have, Theorem 4.4. Under Assumptions (A1)-(A5), there exists δ > such thatfor any C > , one has, for h > small enough Res ( P ) ∩ D h ( δ , C ) = { E k ( h ); k ∈ Z } ∩ D h ( δ , C ) where the E k ( h ) ’s are complex numbers that satisfy, (4.11) E k ( h ) = e k ( h ) + O ( h ) , uniformly as h → + . Remark 4.1. The ellipticity of W assumed in Assumption (A5) is not usedin this theorem. It will be used in the next section in the microlocal methodfor a more precise estimate.Proof. We set, f ( E, h ) := cos A ( E ) h − e i π π h √ W ( E ) . Then, by Proposition 4.3, we have f ( E, h ) = O ( h / ), and the quantizationcondition (4.1) can be written as,(4.12) cos A ( E ) h = f ( E, h ) . We first observe that, near E , the roots of the equation cos( A ( E ) /h ) = 0are precisely given by E = e k ( h ). Moreover, since A (cid:48) ( E ) = (cid:82) ca ( E − V ( x )) − / dx (cid:54) = 0, we see that the distance between two consecutive e k ( h )’sis of order h . As a consequence, if E is at a distance εh from the e k ( h ) withsmall enough positive ε , then cos( A ( E ) /h ) remains at some h -independentpositive distance from 0. Therefore, we can apply the Rouch´e theorem andconclude that, for each k such that e k ( h ) ∈ [ E − δ, E + δ ], and for h > E k ( h ) = e k ( h ) + o ( h ) , and conversely, all the roots of (4.12) in D h ( δ , C ) are of this type. Butsince f = O ( h / ), we immediately see that these roots actually satisfy, E k ( h ) = e k ( h ) + O ( h ) , and the result is proved. (cid:3) Now, in order to specify better the location of the resonances (in particulartheir widths), we will compare the corresponding resonant states with formalconstructions that will be made microlocally (that is, in phase-space) nearthe characteristic set of P − E , Char( P − E ) = Γ ∪ Γ (see Figure 2). , A. MARTINEZ AND T. WATANABE Microlocal constructions near the crossing points of thecharacteristic sets For E ∈ D h ( δ , C ), E := Re E , we plan to construct microlocal solutionsto the (matrix) equation ( P − E ) u = 0, that are concentrated near somearbitrary point of Γ ( E ) ∪ Γ ( E ), where, for j = 1 , 2, we have set,Γ j ( E ) := { ( x, ξ ) ∈ R ; ξ + V j ( x ) = E } . Actually, since we plan to compare these solutions with the resonant statesobtained from Theorem 4.4, we will construct them in such a way that theyare “out-going”, which in this case means that they have no microsupporton the in-coming set Γ − ,R ( E ), defined by,(5.1) Γ − ,R ( E ) := { ( x, ξ ) ∈ R ; ξ + V ( x ) = E , x > , ξ < } . This fact is well-known in the scalar case (see [He Sj], [Be-Ma]), and oursystem is reduced to such a scalar pseudo-differential operator on Γ − ,R thanksto the microlocal ellipticity of P . In addition, since such constructions arestandard away from the crossing points ρ ± ( E ) := (0 , ±√ E ) (they are theusual WKB constructions), we will concentrate on small neighbourhoods of ρ ± ( E ). Figure 3: Characteristic sets on the phase-spaceAt a first stage, we work with real values of E only, considering E as anextra-parameter. At the end, we will just observe that all our constructionsdepend analytically on E , and can be extended to complex values as longas Im E remains O ( h ).For j = 1 , 2, we also set,Γ j,L = { ( x, ξ ) ∈ Γ j ( E ); x < } , Γ ± j,L = { ( x, ξ ) ∈ Γ j ( E ); x < , ± ξ > } , IDTHS OF RESONANCES ABOVE AN ENERGY-LEVEL CROSSING 25 Γ j,R = { ( x, ξ ) ∈ Γ j ( E ); x > } , Γ ± j,R = { ( x, ξ ) ∈ Γ j ( E ); x > , ± ξ > } . We start by working near ρ − ( E ) where we are going to construct a basis ofmicrolocal solutions, with one of them microlocally concentrated on Γ ( E ) ∪ Γ ( E ) \ Γ − ,R ( E ). In all of these constructions, the various pseudodifferentialoperators are considered at a formal level only, which means that they areidentified with their formal analytic Weyl-semiclassical symbol (near a givenpoint of R n ), as in [HeSj2, Appendix]. The same thing is valid for theFourier integral operators, once their phase function is fixed. Moreover,if A is such a formal analytic pseudodifferential operator, with symbol a defined near ( x , ξ ) ∈ R n , one can define its (microlocal) action on D (cid:48) ( R )by re-summing its symbol up to O ( e − α/h ) with α > x , ξ ), and by taking itsusual Weyl-quantization. In that case, the writing(5.2) Av ∼ x , ξ )just means that the microsupport M S ( Av ) of Av does not contain ( x , ξ )(and thus neither a neighbourhood of it). Here, the notion of microsupportis the one used, e.g., in [HeSj2, Ma].In particular, if a vanishes near ( x , ξ ), then (5.2) is valid for any v ∈ L loc ,possibly h -dependent, with its local L -norm near x that is O ( h − N ) forsome N > 0. In that case, we also write A ∼ x , ξ ).Now, microlocally near ρ − ( E ), we investigate the solutions v = ( v , v ) to( P − E ) v ∼ 0. Thanks to Assumption (A5), the operator W ∗ is ellipticat ρ − ( E ). Therefore, by using the symbolic calculus, we can construct amicrolocal parametrix to it, that we denote by W − ∗ . Then, microlocallynear ρ − ( E ), the system ( P − E ) v ∼ v ∼ − h W − ∗ ( P − E ) v ; W ∗ ( P − E ) W − ∗ ( P − E ) v − h W ∗ W v ∼ . Since the principal symbol of Q := W ∗ ( P − E ) W − ∗ ( P − E ) − h W ∗ W is( ξ + V ( x ) − E )( ξ + V ( x ) − E ), it has a saddle point at ρ − ( E ), and we canapply Theorem b.1 of [HeSj2] or, rather, its generalization to non-selfadjointoperators given in [Ra, Proposition 6.1] (see also [Ba, Proposition 4.6]). Weobtain the existence of a formal Fourier integral operator U (with associatedcanonical transform κ sending ρ − ( E ) to (0 , F = F ( t, h ) (defined near t = 0), such that,(5.4) U F ( Q, h ) U − ∼ 12 ( yhD y + hD y · y ) =: G microlocally near (0 , . (Here, the formal pseudodifferential operator F ( Q, h ) is defined as in [HeSj2,Appendix].)In particular, (cid:101) v := U v is a solution to,(5.5) G (cid:101) v ∼ F (0 , h ) (cid:101) v microlocally near (0 , . , A. MARTINEZ AND T. WATANABE Actually, there are several ways of reducing Q to G , depending on whichaffine transformation is used for sending T ρ − ( E ) Γ ( E ) ∪ T ρ − ( E ) Γ ( E ) onto { yη = 0 } . We show, Proposition 5.1. The reduction (5.4) can be made in such a way that onehas, F (0 , h ) = − i h + O ( h ) uniformly with respect to h > small enough and E ∈ D h ( δ , C ) . Proof. Setting τ = V (cid:48) (0) and τ = − V (cid:48) (0) (both positive by Assumption(A4)), we obtain that, near ρ − ( E ), the principal symbol q of Q satisfies, q ( x, ξ ) = 4 E ( ξ + √ E − τ √ E x )( ξ + √ E + τ √ E x ) + O ( | x | + | ξ + √ E | ) . As it can be seen in the proof of [HeSj2, Theoreme b.1], the first step inthe reduction (5.4) consists in transforming q into αyη + O ( | ( y, η ) | ) (with α (cid:54) = 0 constant) by means of an affine canonical transformation κ . Here,we choose κ ( x, ξ ) = ( y, η ) defined by,(5.6) y = τ √ E x − ξ − √ E ; η = τ τ + τ x + 2 √ Eτ + τ ( ξ + √ E ) . One can immediately check that κ is symplectic, and that under this change q ( x, ξ ) becomes − τ + τ ) √ E yη + O ( | ( y, η ) | ). Then, the next steps inthe proof of [HeSj2, Theoreme b.1] mainly consist in correcting this choiceby terms that are O ( | ( y, η ) | ), and in constructing F ( t, h ) in order to finallyobtain (5.4). In particular, the canonical transformation κ associated to U satisfies,(5.7) κ ( x, ξ ) = κ ( x, ξ ) + O ( | x | + | ξ + √ E | ) . Moreover, by the Weyl-symbolic calculus (see [HeSj2]), we also know thatthe Weyl-symbol σ of U F ( Q, h ) U − satisfies, σ ( y, η ; h ) = F ( q ( κ − ( y, η )) , h ) + O ( h ) , where q stands for the Weyl-symbol of Q . Thus, denoting by (cid:80) k ≥ h k f k ( t )the semiclassical asymptotic expansion of F ( t, h ), and q the subprincipalsymbol of Q , by a Taylor expansion we deduce from (5.4),(5.8) f ( q ( κ − ( y, η ))) = yηf ( q ( κ − ( y, η ))) + q ( κ − ( y, η )) f (cid:48) ( q ( κ − ( y, η ))) = 0 . In particular, we have f (0) = 0, and, applying ∂ y ∂ η to the first equationof (5.8) and taking the value at y = η = 0, we also obtain (using that q ( κ − ( y, η )) = − τ + τ ) √ E yη + O ( | ( y, η ) | )),(5.9) − τ + τ ) √ Ef (cid:48) (0) = 1 . IDTHS OF RESONANCES ABOVE AN ENERGY-LEVEL CROSSING 27 Then, observing that q ( x, ξ ) = i { ξ + V ( x ) , ξ + V ( x ) } + O ( | ξ + V ( x ) | + | ξ + V ( x ) | ) = i ξ ( V (cid:48) ( x ) − V (cid:48) ( x )) + O ( | ξ + V ( x ) | + | ξ + V ( x ) | ) (so that q ( κ − (0 , − i √ E ( τ + τ )), we deduce, f (0) = i √ E ( τ + τ ) f (cid:48) (0) = − i , and the result follows. (cid:3) Remark 5.2. If in (5.6) we had exchanged the roles of y and η (e.g. by thesymplectic change ( y, η ) (cid:55)→ ( η, − y ) ), we would have obtained F (0 , h ) = i h + O ( h ) and the equation (5.5) would have become yd y (cid:101) v = − (1 + O ( h )) (cid:101) v ,making the construction of the solution less easy (as we will see, with ourchoice, instead, (5.5) becomes yd y (cid:101) v = O ( h ) (cid:101) v ). In that case, we also have, Proposition 5.3. It holds that F (0 , h ) = − i h + µh , with µ = µ ( h ) = − r (0) + r (0) E τ + τ ) √ E + O ( h ) uniformly with respect to h > small enough and E ∈ D h ( δ , C ) . Proof. We first observe that, under the conjugation by U , the characteristicset of P − E is changed into { η = 0 } . In particular, it projects bijectively onthe base R y near (0 , P − E ) u = 0near ρ − ( E ) are transformed into smooth functions of y . If we set v = h − / u − ,L (so that v (0) is O (1)), then v is solution to Qv = − h W ∗ W v , andthe smooth function (cid:101) v := U u − ,L is solution to,(5.10) G (cid:101) v ∼ U F ( − h W ∗ W, h ) U − (cid:101) v microlocally near (0 , . Writing F ( t, h ) = F (0 , h ) + tF ( t, h ) = F (0 , h ) + t ( (cid:101) F ( t ) + O ( h )), we deduce,(5.11) (cid:18) yhD y − ih (cid:19) (cid:101) v ∼ F (0 , h ) (cid:101) v − h (cid:101) F (0) U W ∗ W U − (cid:101) v + h R (cid:101) v, microlocally near (0 , R is a 0-th order semiclassical pseudodiffer-ential operator. Applying the FBI transform T : w (cid:55)→ T w given by, T w ( y, η ; h ) := h − (cid:90) e i ( y − y (cid:48) ) η/h − ( y − y (cid:48) ) / h χ ( y (cid:48) ) w ( y (cid:48) ) dy (cid:48) , (where χ is a cut-off function near 0), integrating by parts, and taking thevalue at y = η = 0, we deduce from (5.11),(5.12) − ih T (cid:101) v (0 , 0) = (cid:16) F (0 , h ) − h (cid:101) F (0) | W ( ρ − ( E )) | (cid:17) T (cid:101) v (0 , 0) + O ( h ) , where we have denoted by W ( x, ξ ) := r ( x ) + ir ( x ) ξ the principal symbolof W ( x, hD x ).Since the microsupport of (cid:101) v near (0 , 0) coincides with η = 0, and (cid:101) v (0) ∼ , A. MARTINEZ AND T. WATANABE a stationary-phase expansion shows that T (cid:101) v (0 , ∼ 1, too, as h → + .Therefore (5.12) implies,(5.13) F (0 , h ) = − ih h (cid:101) F (0) | W ( ρ − ( E )) | + O ( h ) . On the other hand, by definition we have, (cid:101) F (0) = F (0 , h ) + O ( h ) = ∂ t F (0 , h ) + O ( h ) = f (cid:48) (0) + O ( h ) , and the result follows from (5.9) and (5.13). (cid:3) Now, thanks to Proposition 5.3, we can re-write (5.5) as,(5.14) y d (cid:101) v dy ∼ iµh (cid:101) v microlocally near (0 , . A basis of solutions to (5.14) is given by the two functions, u (cid:97) ( y ) := H ( − y ) | y | iµh ; u (cid:96) ( y ) := H ( y ) | y | iµh , where H stands for the Heaviside function. In particular, near (0 , M S ( u (cid:97) ( y )) = { y < , η = 0 } ∪ { y = 0 } ; M S ( u (cid:96) ( y )) = { y > , η = 0 } ∪ { y = 0 } . As a consequence, by construction the functions v (cid:97) := U − u (cid:97) and v (cid:96) := U − u (cid:96) are solutions to Qv ∼ V − of ρ − ( E ), and their microsupports satisfy, M S ( v (cid:97) ( x )) ∩ V − ⊂ (Γ ( E ) ∪ Γ − ,L ( E )) ∩ V − ; M S ( v (cid:96) ( x )) ∩ V − ⊂ (Γ ( E ) ∪ Γ − ,R ( E )) ∩ V − . Going back to (5.3), if we set v (cid:97) := − h W − ∗ ( P − E ) v (cid:97) , v (cid:96) := − h W − ∗ ( P − E ) v (cid:96) , v (cid:97) := ( v (cid:97) , v (cid:97) ), and v (cid:96) = ( v (cid:96) , v (cid:96) ), then v (cid:97) and v (cid:96) are both solutionsto ( P − E ) v ∼ V − , and their microsupports satisfy, M S ( v (cid:97) ( x )) ∩ V − ⊂ (Γ ( E ) ∪ Γ − ,L ( E )) ∩ V − ; M S ( v (cid:96) ( x )) ∩ V − ⊂ (Γ ( E ) ∪ Γ − ,R ( E )) ∩ V − . Then we plan to compute the microlocal asymptotic behaviour of v (cid:97) and v (cid:96) on each of the three branches of their microsupport.We first observe that, by a convenient normalization of the Fourier integraloperator U , we can assume, U − u ( x ; h ) = (cid:90) R e iψ ( x,y ) /h c ( x, y ; h ) u ( y ) dy, where c ∼ (cid:80) k ≥ h k c k is an analytic symbol, c (0 , 0) = 1, and ψ is a generat-ing function of κ − , in the sense that we have κ − : ( y, −∇ y ψ ) (cid:55)→ ( x, ∇ x ψ ).Now we compute the asymptotic behaviour of the solutions v (cid:97) and v (cid:96) oneach of Γ − ,L ( E ), Γ − ,R ( E ), Γ − ,R ( E ), or Γ − ,L ( E ). Microlocally near thesecurves, the operator Q is of principal type (it can be microlocally trans-formed into hD y ), and thus the space of microlocal solutions to ( P − E ) u = 0 IDTHS OF RESONANCES ABOVE AN ENERGY-LEVEL CROSSING 29 is one-dimensional. In addition, using the phase functions − ν j ( x ) ( j = 1 , a − ( x ; h ) e − iν ( x ) /h on Γ − ,R ( E ) or Γ − ,L ( E ), and of the type b − ( x ; h ) e − iν ( x ) /h on Γ − ,R ( E ) or Γ − ,L ( E ), where a and b are elliptic analytic (vector-valued)symbols that become singular at x = 0. More precisely, Proposition 5.4. There exist functions f − ,L , f − ,R , f − ,L , f − ,R ∈ L ( R ) suchthat ( P − E ) f − ,L ∼ , f − ,L ∼ (cid:32) a − ha − (cid:33) e − iν ( x ) /h microlocally on Γ − ,L ( E ) , ( P − E ) f − ,R ∼ , f − ,R ∼ (cid:32) a − ha − (cid:33) e − iν ( x ) /h microlocally on Γ − ,R ( E ) , ( P − E ) f − ,L ∼ , f − ,L ∼ (cid:32) hb − b − (cid:33) e − iν ( x ) /h microlocally on Γ − ,L ( E ) , ( P − E ) f − ,R ∼ , f − ,R ∼ (cid:32) hb − b − (cid:33) e − iν ( x ) /h microlocally on Γ − ,R ( E ) . Here, ν j ( x ) := (cid:90) x (cid:113) E − V j ( t ) dt ( j = 1 , , and a − j = a − j ( x ; h ) ∼ (cid:80) k ≥ h k a − j,k ( x ) and b − j = b − j ( x ; h ) ∼ (cid:80) k ≥ h k b − j,k ( x ) ( j = 1 , ) are analytic symbols whose first coefficients are given by (5.15) , (5.16) below.Proof. On Γ − ( E ), writing a − = (cid:18) a − ha − (cid:19) with a − j ( x ; h ) ∼ (cid:80) k ≥ h k a − j,k ( x )( j = 1 , a − , = 1( E − V ) ; a − , = r + ir √ E − V ( V − V )( E − V ) . Similarly, on Γ − ( E ), writing b − = (cid:18) hb − b − (cid:19) with b − j ∼ (cid:80) k ≥ h k b − j,k ( x )( j = 1 , b − , = 1( E − V ) ; b − , = r − ir √ E − V ( V − V )( E − V ) . (cid:3) Proposition 5.5. Let V − be a small enough neighbourhood of ρ − ( E ) . Thefunctions v (cid:97) and v (cid:96) have the following microlocal asymptotic behaviours: (5.17) v (cid:97) ∼ α (cid:97) R h iµh f − ,R microlocally on Γ − ,R ( E ) ∩ V − ; v (cid:97) ∼ α (cid:97) L h iµh f − ,L microlocally on Γ − ,L ( E ) ∩ V − ; v (cid:97) ∼ β (cid:97) L √ hf − ,L microlocally on Γ − ,L ( E ) ∩ V − , , A. MARTINEZ AND T. WATANABE and, (5.18) v (cid:96) ∼ α (cid:96) R h iµh f − ,R microlocally on Γ − ,R ( E ) ∩ V − ; v (cid:96) ∼ α (cid:96) L h iµh f − ,L microlocally on Γ − ,L ( E ) ∩ V − ; v (cid:96) ∼ β (cid:96) R √ hf − ,R microlocally on Γ − ,R ( E ) ∩ V − , with α (cid:97) S = α (cid:97) S ( h ) ∼ (cid:80) k ≥ h k α (cid:97) S,k , β (cid:97) S = β (cid:97) S ( h ) ∼ (cid:80) k ≥ h k β (cid:97) S,k , α (cid:96) S = α (cid:96) S ( h ) ∼ (cid:80) k ≥ h k α (cid:96) S,k , β (cid:96) S = β (cid:96) S ( h ) ∼ (cid:80) k ≥ h k β (cid:96) S,k ( S = L, R ), (5.19) α (cid:97) L, = α (cid:97) R, = − α (cid:96) L, = − α (cid:96) R, = i ( τ + τ ) E r (0) + ir (0) √ E ; β (cid:97) L, = β (cid:96) R, = (cid:112) π ( τ + τ ) e iπ/ . Proof. We just have to determine the coefficients of microlocal proportion-ality between v (cid:97) (respectively v (cid:96) ) and the WKB solutions.Using (5.6)-(5.7), we see that the phase function ψ in the definition of U − satisfies,(5.20) ψ ( x, y ) = τ √ E x + √ Eτ + τ y − xy − √ E x + O ( | ( x, y ) | ) . Near 0 the map y (cid:55)→ ψ ( x, y ) admits a unique critical point y c ( x ), that isnon-degenerate and is such that, y c ( x ) = τ + τ √ E x + O ( x ) . In particular, the corresponding critical value ψ ( x, y c ( x )) vanishes at x = 0,and since we know that κ − sends { η = 0 } onto Γ ( E ) ∩ V − , we necessarilyhave ψ ( x, y c ( x )) = − ν ( x ) . Working with v (cid:97) on x < 0, by definition we have, v (cid:97) ( x ; h ) = (cid:90) −∞ e iψ ( x,y ) /h c ( x, y ; h ) | y | iµh dy. Only two points contribute, up to exponentially small quantities, to thisintegral. One is the critical point y c ( x ) of ψ , and the other one is the singularpoint y = 0 of u (cid:97) . By the stationary-phase theorem, the contribution of y c ( x ) is of the type √ πh σ (cid:97) ( x ; h ) e − iν ( x ) /h , where σ (cid:97) ( x ; h ) ∼ (cid:80) k ≥ h k σ (cid:97) k ( x ) is a classical symbol of order 0, with σ (cid:97) ( x ) = e iπ/ ( ∂ y ψ ( x, y c ( x ))) − c ( x, y c ( x )) (here we use the fact that ∂ y ψ (0 , 0) = √ Eτ + τ > y = 0 can be computed by a complexchange of contour of integration, that reduces the integral to a Laplacetransform (see, e.g., [Er]). After the change of scale t → ht , we obtain anexpression of the type, h iµh (cid:101) σ (cid:97) ( x ; h ) e iψ ( x, /h IDTHS OF RESONANCES ABOVE AN ENERGY-LEVEL CROSSING 31 where (cid:101) σ (cid:97) ( x ; h ) ∼ (cid:80) k ≥ h k (cid:101) σ (cid:97) k ( x ) is a classical symbol of order 0, with (cid:101) σ (cid:97) ( x ) = ix − c ( x, κ − sends { y = 0 } onto Γ ( E ) ∩ V − and ψ (0 , 0) = 0, wenecessarily have ψ ( x, 0) = − ν ( x ), and thus, locally on { x < } , we finallyobtain, v (cid:97) ( x ; h ) = √ πh σ (cid:97) ( x, h ) e − iν ( x ) /h + h iµh (cid:101) σ (cid:97) ( x ; h ) e − iν ( x ) /h + O ( e − ε/h ) , with ε > v (cid:97) there, and thus also the microlocal asymptotic behaviourof v (cid:97) on Γ − ,L ( E ) and on Γ − ,L ( E ). Comparing with the microlocal WKBsolutions we deduce the existence of constants α (cid:97) L ∼ (cid:80) k ≥ h k α (cid:97) L,k and β (cid:97) L ∼ (cid:80) k ≥ h k β (cid:97) L,k , such that (5.17) holds, and thus in particular, √ π σ (cid:97) = β (cid:97) L b − ; (cid:101) σ (cid:97) = α (cid:97) L a − . At the level of principal symbols, we obtain, √ π e iπ/ ( ∂ y ψ ( x, y c ( x ))) − c ( x, y c ( x )) = β (cid:97) L, ( E − V ( x )) − ; ix − c ( x, 0) = α (cid:97) L, r ( x ) + ir ( x ) (cid:112) E − V ( x )( V ( x ) − V ( x ))( E − V ( x )) , where, at a first stage, the equalities are valid for x < V ( x ) − V ( x ) = ( τ + τ ) x + O ( x ) , we can multiply the second equality by x and make x tend to 0. Then, using (5.20) and the fact that c (0 , 0) = 1,we obtain the values of α (cid:97) L, and β (cid:97) L, given in (5.19).The same arguments hold for x > 0, with the difference that only y = 0contributes to the integral (in this case the critical point does not belongto the interval of integration), and we obtain the expression of α (cid:97) R given in(5.19).The result for v (cid:96) follows in the same way. (cid:3) Now we exchange the role of the indices 1 and 2. Namely, we reduce thesystem to a scalar equation with unknown v :(5.21) v ∼ − h W − ( P − E ) v ; (cid:98) Qv := W ( P − E ) W − ( P − E ) v − h W W ∗ v ∼ . The equation (cid:98) Qv ∼ G (cid:98) v ∼ (cid:98) F (0 , h ) (cid:98) v with (cid:98) v := (cid:98) U v , wherethe Fourier integral operator (cid:98) U is given by, (cid:98) U − u ( x, h ) = (cid:90) R e i (cid:98) ψ ( x,y ) /h (cid:98) c ( x, y ; h ) u ( y ) dy, (cid:98) ψ ( x, y ) = − τ √ E x − √ Eτ + τ y + xy − √ Ex + O ( | ( x, y ) | ) , (cid:98) c ∼ ∞ (cid:88) k =0 h k c k ( x, y ) , (cid:98) c (0 , 0) = 1 . , A. MARTINEZ AND T. WATANABE and the formal analytic symbol (cid:98) F ( t, h ) is such that, (cid:98) F (0 , h ) = − i h + (cid:98) µh , with (cid:98) µ = + r (0) + r (0) E τ + τ ) √ E + O ( h ) . Notice that (cid:98) µ = − µ modulo O ( h ). Let us define as before (cid:98) u (cid:97) ( y ) := H ( − y ) | y | i (cid:98) µh , (cid:98) u (cid:96) ( y ) := H ( y ) | y | i (cid:98) µh , (cid:98) v (cid:97) := (cid:98) U − (cid:98) u (cid:97) , (cid:98) v (cid:96) := (cid:98) U − (cid:98) u (cid:96) , (cid:98) v (cid:97) = − h W − ( P − E ) (cid:98) v (cid:97) , (cid:98) v (cid:96) = − h W − ( P − E ) (cid:98) v (cid:96) . Then (cid:98) v (cid:97) := ( (cid:98) v (cid:97) , (cid:98) v (cid:97) ) and (cid:98) v (cid:96) := ( (cid:98) v (cid:96) , (cid:98) v (cid:96) ) are both solutions to ( P − E ) v ∼ ρ − ( E ) that we still denote by V − ,and their microsupports satisfy, M S ( (cid:98) v (cid:97) ) ∩ V − ⊂ (Γ ( E ) ∪ Γ − ,L ( E )) ∩ V − ; M S ( (cid:98) v (cid:96) ) ∩ V − ⊂ (Γ ( E ) ∪ Γ − ,R ( E )) ∩ V − . Proposition 5.6. The functions (cid:98) v (cid:97) and (cid:98) v (cid:96) have the following microlocalasymptotic behaviours: (5.22) (cid:98) v (cid:97) ∼ (cid:98) α (cid:97) R h i ˆ µh f − ,R microlocally on Γ − ,R ( E ) ∩ V − ; (cid:98) v (cid:97) ∼ (cid:98) α (cid:97) L h i ˆ µh f − ,L microlocally on Γ − ,L ( E ) ∩ V − ; (cid:98) v (cid:97) ∼ (cid:98) β (cid:97) L √ hf − ,L microlocally on Γ − ,L ( E ) ∩ V − , and, (5.23) (cid:98) v (cid:96) ∼ (cid:98) α (cid:96) R h i ˆ µh f − ,R microlocally on Γ − ,R ( E ) ∩ V − ; (cid:98) v (cid:96) ∼ (cid:98) α (cid:96) L h i ˆ µh f − ,L microlocally on Γ − ,L ( E ) ∩ V − ; (cid:98) v (cid:96) ∼ (cid:98) β (cid:96) R √ hf − ,R microlocally on Γ − ,R ( E ) ∩ V − , with (cid:98) α (cid:97) S = (cid:98) α (cid:97) S ( h ) ∼ (cid:80) k ≥ h k (cid:98) α (cid:97) S,k , (cid:98) β (cid:97) S = (cid:98) β (cid:97) S ( h ) ∼ (cid:80) k ≥ h k (cid:98) β (cid:97) S,k , (cid:98) α (cid:96) S = (cid:98) α (cid:96) S ( h ) ∼ (cid:80) k ≥ h k (cid:98) α (cid:96) S,k , (cid:98) β (cid:96) S = (cid:98) β (cid:96) S ( h ) ∼ (cid:80) k ≥ h k (cid:98) β (cid:96) S,k ( S = L, R ), (5.24) + (cid:98) α (cid:97) L, = + (cid:98) α (cid:97) R, = − (cid:98) α (cid:96) L, = − (cid:98) α (cid:96) R, = i ( τ + τ ) E r (0) − ir (0) √ E ; (cid:98) β (cid:97) L, = (cid:98) β (cid:96) R, = (cid:112) π ( τ + τ ) e − iπ/ . From Proposition 5.5 and Proposition 5.6, we can compute the followingtransfer matrix, which connects microlocal data on Γ − ,L and Γ − ,R to thoseon Γ − ,R and Γ − ,L . Proposition 5.7. Suppose u ( x, h ) ∈ L ( R ) satisfies ( P − E ) u ∼ microlo-cally in a neighbourhood V − of ρ − ( E ) and set u ∼ t − ,L f − ,L microlocally on Γ − ,L ( E ) ,u ∼ t − ,R f − ,R microlocally on Γ − ,R ( E ) , IDTHS OF RESONANCES ABOVE AN ENERGY-LEVEL CROSSING 33 u ∼ t − ,L f − ,L microlocally on Γ − ,L ( E ) ,u ∼ t − ,R f − ,R microlocally on Γ − ,R ( E ) , for constants t − j,S = t − j,S ( E, h ) . Then it holds that (5.25) (cid:32) t − ,R t − ,L (cid:33) = (cid:32) τ − , ( E, h ) τ − , ( E, h ) τ − , ( E, h ) τ − , ( E, h ) (cid:33) (cid:32) t − ,L t − ,R (cid:33) , where the coefficients τ − j,k have the asymptotic behaviour τ − , = 1 + O ( h ) , τ − , = e π i τ − h + iµh + O ( h ) ,τ − , = e − π i τ + h − iµh + O ( h ) , τ − , = 1 + O ( h ) , as h → + uniformly for E ∈ D h ( δ , C ) . Here τ ± are constants defined by τ ± = (cid:114) πτ + τ (cid:16) r (0) E − ± ir (0) E (cid:17) . Proof. If t − ,L = 1 and t − ,R = 0, then u should be equal to ( α (cid:97) L h iµh ) − v (cid:97) microlocally in a neighbourhood of ρ − , and hence we have τ − , = t − ,R = α (cid:97) R α (cid:97) L , τ − , = t − ,L = β (cid:97) L α (cid:97) L h − iµh . Similarly, if t − ,L = 0 and t − ,R = 1, then u should be equal to ( (cid:98) α (cid:96) R h i (cid:98) µh ) − (cid:98) v (cid:96) microlocally in a neighbourhood of ρ − , and hence we have τ − , = t − ,R = (cid:98) β (cid:96) R (cid:98) α (cid:96) R h − i (cid:98) µh , τ − , = t − ,L = (cid:98) α (cid:96) L (cid:98) α (cid:96) R . Then Proposition 5.7 follows from (5.19) and (5.24). (cid:3) We obtain in the same way the transfer matrix near ρ + ( E ). First, we defineWKB solutions microlocally on the characteristics Γ + j,S as Proposition 5.4: Proposition 5.8. There exist functions f +1 ,L , f +1 ,R , f +2 ,L , f +2 ,R ∈ L ( R ) suchthat ( P − E ) f +1 ,L ∼ , f +1 ,L ∼ (cid:32) a +1 ha +2 (cid:33) e iν ( x ) /h microlocally on Γ +1 ,L ( E ) , ( P − E ) f +1 ,R ∼ , f +1 ,R ∼ (cid:32) a +1 ha +2 (cid:33) e iν ( x ) /h microlocally on Γ +1 ,R ( E ) , ( P − E ) f +2 ,L ∼ , f +2 ,L ∼ (cid:32) hb +1 b +2 (cid:33) e iν ( x ) /h microlocally on Γ +2 ,L ( E ) , ( P − E ) f +2 ,R ∼ , f +2 ,R ∼ (cid:32) hb +1 b +2 (cid:33) e iν ( x ) /h microlocally on Γ +2 ,R ( E ) . , A. MARTINEZ AND T. WATANABE Here, a + j = a − j ( x ; h ) ∼ (cid:80) k ≥ h k a + j,k ( x ) and b + j = b + j ( x ; h ) ∼ (cid:80) k ≥ h k b + j,k ( x ) ( j = 1 , ) are analytic symbols whose first coefficients are given by (5.26) a +1 , = 1( E − V ) ; a +2 , = r − ir √ E − V ( V − V )( E − V ) . (5.27) b +2 , = 1( E − V ) ; b +1 , = r + ir √ E − V ( V − V )( E − V ) . Proposition 5.9. Suppose u ( x, h ) ∈ L ( R ) satisfies ( P − E ) u ∼ microlo-cally in a neighbourhood V + of ρ + ( E ) and set u ∼ t +1 ,L f +1 ,L microlocally on Γ +1 ,L ( E ) ,u ∼ t +1 ,R f +1 ,R microlocally on Γ +1 ,R ( E ) ,u ∼ t +2 ,L f +2 ,L microlocally on Γ +2 ,L ( E ) ,u ∼ t +2 ,R f +2 ,R microlocally on Γ +2 ,R ( E ) . Then it holds that (5.28) (cid:32) t +1 ,L t +2 ,R (cid:33) = (cid:32) τ +1 , ( E, h ) τ +1 , ( E, h ) τ +2 , ( E, h ) τ +2 , ( E, h ) (cid:33) (cid:32) t +1 ,R t +2 ,L (cid:33) , where τ +1 , = 1 + O ( h ) , τ +1 , = − e − π i τ + h − iµh + O ( h ) ,τ +2 , = − e π i τ − h + iµh + O ( h ) , τ +2 , = 1 + O ( h ) , as h → + uniformly for E ∈ D h ( δ , C ) . Microlocal connection formulas and monodromy condition In the previous section, we defined WKB solutions f ± j,S ( x, h ) microlocallyalong the 8 curves Γ ± j,S on Γ ( E ) ∪ Γ ( E ) divided by the 2 crossing points ρ ± ( E ) and 3 turning points (caustics) ( a ( E ) , , ( b ( E ) , 0) and ( c ( E ) , P − E ) u = 0, itis necessary to know the connection formulae between these WKB solutions.The connection formulae near the crossing points were given in Proposition5.7 and 5.9, and it remains to have those between these microlocal WKBsolutions at the turning points. The following lemma is essentially due toMaslov. Lemma 6.1. Let u satisfy ( P − E ) u ∼ microlocally near Γ ,S , and suppose u ∼ t +1 ,S f +1 ,S microlocally on Γ +1 ,S ( E ) ,u ∼ t − ,S f − ,S microlocally on Γ − ,S ( E ) , for S = L, R . Then it holds that (6.1) t +1 ,S = σ ,S t − ,S , with constants σ ,S , which behave, as h → + , σ ,L = − ie +2 iS ,L /h + O ( h ) , σ ,R = ie − iS ,R /h + O ( h ) . IDTHS OF RESONANCES ABOVE AN ENERGY-LEVEL CROSSING 35 Here S ,S ( E ) are the action integrals defined by S ,L ( E ) := (cid:90) a ( E ) (cid:112) E − V ( x ) dx, S ,R ( E ) := (cid:90) c ( E )0 (cid:112) E − V ( x ) dx. Similarly, let u be a solution to ( P − E ) u ∼ microlocally near Γ ,L , andsuppose u ∼ t +2 ,L f +2 ,L microlocally on Γ +2 ,L ( E ) ,u ∼ t − ,L f − ,L microlocally on Γ − ,L ( E ) . Then it holds that (6.2) t +2 ,L = σ ,L t − ,L , with σ ,L = − ie +2 iS ,L /h + O ( h ) , where S ,L ( E ) := (cid:90) b ( E ) (cid:112) E − V ( x ) dx. Proof. We compute σ ,L here. The computations of σ ,R and σ ,L are similar.Microlocally near ( x, ξ ) = ( a ( E ) , P − E is elliptic andinvertible. Hence the system (2.1) is reduced to a single semiclassical pseudo-differential equation for u :( P − E ) u − h Ru = 0 , where R = W ( P − E ) − W ∗ . The idea of Maslov is to represent a solution u as Fourier inverse transform of a WKB function e ig ( ξ ) /h c ( ξ, h ), c ( ξ, h ) ∼ (cid:80) c k ( ξ ) h k in the momentum variable, that is, u ( x, h ) = 1 √ πh (cid:90) R e i (˜ xξ + g ( ξ )) /h c ( ξ, h ) dξ, where ˜ x = x − a ( E ). The WKB function e ig ( ξ ) /h c ( ξ, h ) is a solution to thesemiclassical pseudo-differential equation(6.3) Q ( ξ, hD ξ ; h )( e ig ( ξ ) /h c ( ξ, h )) = 0 , where Q ( ξ, hD ξ ; h ) is the standard quantization of q ( ξ, ξ ∗ ; h ) given by q ( ξ, ξ ∗ ; h ) = p ( − ξ ∗ + a ( E ) , ξ ) − E − h r ( − ξ ∗ + a ( E ) , ξ ; h ) , where ξ ∗ is the dual variable of ξ and r ( x, ξ ; h ) is the standard symbol of R . In order to have the phase function and the asymptotic expansion of thesymbol, we write e − ig ( ξ ) /h Q ( ξ, hD ξ ; h ) (cid:16) e ig ( ξ ) /h c ( ξ, h ) (cid:17) = 12 πh (cid:90) (cid:90) e i { ( ξ − η ) ξ ∗ − ( g ( ξ ) − g ( η )) } /h q ( ξ, ξ ∗ ; h ) c ( η, h ) dηdξ ∗ = 12 πh (cid:90) (cid:90) e i ( ξ − η ) ξ ∗ /h q ( ξ, ξ ∗ + ˜ g ( ξ, η ); h ) c ( η, h ) dηdξ ∗ = 12 πh (cid:90) (cid:90) e i ( ξ − η ) ξ ∗ /h q ( ξ, ξ ∗ + ˜ g ( ξ, η ); h ) c ( η ) dηdξ ∗ + O ( h ) , A. MARTINEZ AND T. WATANABE where ˜ g ( ξ, η ) is defined by g ( ξ ) − g ( η ) = ( ξ − η )˜ g ( ξ, η ). Since ˜ g ( ξ, ξ ) = g (cid:48) ( ξ ), ∂ η ˜ g ( ξ, ξ ) = g (cid:48)(cid:48) ( ξ ) and q ( ξ, ξ ∗ + ˜ g ( ξ, η ); h ) = ξ + V ( a ( E ) − ˜ g ( ξ, η )) − E − ξ ∗ V (cid:48) ( a ( E ) − ˜ g ( ξ, η )) + O (( ξ ∗ ) ) + O ( h ), the last integral in the previousidentities is equal to( ξ + V ( a ( E ) − g (cid:48) ( ξ )) − E ) c ( ξ )+ hi ∂ η (cid:0) V (cid:48) ( a ( E ) − ˜ g ( ξ, η )) c ( η ) (cid:1) | η = ξ + O ( h ) . Therefore we have e − ig ( ξ ) /h Q ( ξ, hD ξ ; h ) (cid:16) e ig ( ξ ) /h c ( ξ, h ) (cid:17) =( ξ + V ( a ( E ) − g (cid:48) ( ξ )) − E ) c ( ξ )+ hi (cid:18) V (cid:48) ( a ( E ) − g (cid:48) ( ξ )) c (cid:48) ( ξ ) − g (cid:48)(cid:48) ( ξ ) V (cid:48)(cid:48) ( a ( E ) − g (cid:48) ( ξ )) c ( ξ ) (cid:19) + O ( h ) . Hence the phase function g ( ξ ) and the coefficient c ( ξ ) of the symbol shouldsatisfy respectively the eikonal equation(6.4) ξ + V ( a ( E ) − g (cid:48) ( ξ )) − E = 0 , and the first transport equation(6.5) V (cid:48) ( a ( E ) − g (cid:48) ( ξ )) c (cid:48) ( ξ ) − g (cid:48)(cid:48) ( ξ ) V (cid:48)(cid:48) ( a ( E ) − g (cid:48) ( ξ )) c ( ξ ) = 0 . Recall that, near x = a ( E ), V ( x ) behaves like V ( x ) = V ( a ( E ) + ˜ x ) = E − τ ( E )˜ x + O (˜ x ) , where τ ( E ) is an analytic function near E = E with τ ( E ) > 0. Thismeans that the solution g ( ξ ) of the eikonal equation (6.4) with g (0) = 0satisfies g ( ξ ) = − τ ( E ) ξ + O ( ξ ) , and that there are two real critical points of the phase ˜ xξ + g ( ξ ) for positive˜ x , which are denoted by ξ ± (˜ x ) satisfying ξ ± (˜ x ) = ± (cid:112) τ ( E )˜ x + O (˜ x / ).The critical values ˜ xξ ± (˜ x ) + g ( ξ ± (˜ x )) behave like˜ xξ ± (˜ x ) + g ( ξ ± (˜ x )) = ± (cid:112) τ ( E )˜ x + O (˜ x ) , as ˜ x → 0. Notice that ∓ g (cid:48)(cid:48) ( ξ ± (˜ x )) ∼ ± ξ ± (˜ x ) /τ ( E ) > x > u as h → + near ˜ x positive and close to 0. Since the phase ν ( x ) of the WKB solution f +1 ,L also behaves like − S ,L ( E ) + 23 (cid:112) τ ( E )˜ x + O (˜ x )as ˜ x → 0, it should coincide with ˜ xξ + (˜ x ) + g ( ξ + (˜ x )) except the phase shift − S ,L ( E ). Hence the contribution of the stationary phase from the criticalpoint ξ + (˜ x ) e ih (˜ xξ + (˜ x )+ g ( ξ + (˜ x ))) ∞ (cid:88) k =0 ˜ c + k ( x ) h k , ˜ c +0 ( x ) = e − πi/ c ( ξ + (˜ x )) (cid:112) | g (cid:48)(cid:48) ( ξ + (˜ x )) | IDTHS OF RESONANCES ABOVE AN ENERGY-LEVEL CROSSING 37 should coincide with the WKB expansion t +1 ,L a +1 e iν ( x ) /h microlocally onΓ +1 ,L ( E ), so that one sees e ih S ,L ˜ c +0 = t +1 ,L a +1 . Similarly the contributionfrom the critical point ξ − (˜ x ) e ih (˜ xξ − (˜ x )+ g ( ξ − (˜ x ))) ∞ (cid:88) k =0 ˜ c − k ( x ) h k , ˜ c − ( x ) = e πi/ c ( ξ − (˜ x )) (cid:112) g (cid:48)(cid:48) ( ξ − (˜ x ))should coincide with the WKB expansion t − ,L a − e − iν ( x ) /h microlocally onΓ − ,L ( E ), so that one has e − ih S ,L ˜ c − = t − ,L a − . To compute c ± ( x ), we solvethe first transport equation (6.5). In fact, putting φ ( ξ ) := V (cid:48) ( a − g (cid:48) ( ξ )), wesee that (6.5) is equivalent to (cid:16)(cid:112) φ ( ξ ) c ( ξ ) (cid:17) (cid:48) = 0, so that we get explicitly c ( ξ ) = (cid:0) − V (cid:48) ( a − g (cid:48) ( ξ )) (cid:1) − . Hence the following holds for small ˜ x > c ( ξ ± (˜ x )) = (cid:18) − V (cid:48) ( a + 1 τ ( E ) ξ ± (˜ x ) ) (cid:19) − (1 + O (˜ x )) . The above calculations and the fact that a ± , are independent of the sign ± imply that ˜ c +0 ( x ) / ˜ c − ( x ) = − i + O (˜ x ). Therefore we have verified σ ,L = − ie +2 iS ,L /h + O ( h ). (cid:3) Let E be a resonance in D h ( δ , C ) and u ( x, h ) a resonant state correspond-ing to E . We normalize u in such a way that(6.6) u ∼ f − ,L microlocally on Γ − ,L ( E ) . Since u is a resonant state, we also have(6.7) u ∼ − ,R ( E ) . Then using the microlocal connection formulae of the previous and thissections, we obtain Proposition 6.1. Let u be a resonant state corresponding to a resonance E in D h ( δ , C ) satisfying (6.6). Then it holds that (6.8) u ∼ t +2 ,R f +2 ,R microlocally on Γ +2 ,R ( E ) , with (6.9) t +2 ,R = − i (cid:114) πhτ + τ e ih ( − S ,R + S ,L ) × (cid:20) r (0) E − sin (cid:18) B ( E ) h + π (cid:19) + r (0) E cos (cid:18) B ( E ) h + π (cid:19)(cid:21)(cid:18) O ( h log 1 h ) (cid:19) , where B ( E ) := (cid:90) b ( E ) (cid:112) E − V ( x ) dx + (cid:90) c ( E )0 (cid:112) E − V ( x ) dx. In particular, the leading term of | t +2 ,R | is equal to (6.10) 4 πhτ + τ (cid:12)(cid:12)(cid:12)(cid:12) r (0) E − sin (cid:18) B ( E ) h + π (cid:19) + r (0) E cos (cid:18) B ( E ) h + π (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) . , A. MARTINEZ AND T. WATANABE Proof. Using the notations in Propositions 5.7, 5.9 and Lemma 6.1, we have t +2 ,R = σ ,R τ +2 , τ − , + σ ,L τ +2 , τ − , , = − (cid:114) πhτ + τ (cid:16) e − iθ ( a − ib ) + e iθ ( a + ib ) (cid:17) , where θ = h ( S ,R + S ,L ) − π − µh log h , a = r (0) E − / and b = r (0) E / .A simple computation gives t +2 ,R = − i (cid:114) πhτ + τ e ih ( − S ,R + S ,L ) ( a cos θ − b sin θ ) , = − i (cid:114) πhτ + τ e ih ( − S ,R + S ,L ) (cid:16) a sin( θ + π b cos( θ + π (cid:17) . Hence we obtain (6.9) and (6.10). (cid:3) Similarly, after having made a whole loop on Γ ( E ), we also obtain,(6.11) t − ,L = − e i A ( E ) /h + O ( h ) , and therefore, we have, Proposition 6.2. Any resonance E in D h ( δ , C ) satisfies (6.12) e i A ( E ) /h = − O ( h ) . Precise asymptotics of resonances We have already given in Theorem 4.4 the semiclassical distribution of res-onances modulo O ( h ). In this section, we give a more precise asymptoticbehaviour of the distribution of resonances using the microlocal results ofPropositions 6.1 and 6.2.First, the formula (2.4) about the real part of resonances is a direct conse-quence of Proposition 6.2.Second, the formula (2.5) about the imaginary part of resonances is deducedfrom Proposition 6.1 and the following propositions 7.1 and 7.2. Proposition 7.1. Let E and u be as in Proposition 6.1. Then one has, forany x > c , − Im E = | t +2 ,R | (cid:107) u (cid:107) L ( −∞ ,x ) h (1 + O ( h )) . Proof. Let x be any real number. The width of a resonance E can beexpressed in terms of the resonant state u ( x, h ) = t ( u , u ) and its derivativeat the point x by the formula,(7.1) (Im E ) (cid:107) u (cid:107) L ( −∞ ,x ) = − h Im (cid:16) u (cid:48) ( x ) u ( x ) + u (cid:48) ( x ) u ( x ) (cid:17) + h Im r ( x ) u ( x ) u ( x ) . IDTHS OF RESONANCES ABOVE AN ENERGY-LEVEL CROSSING 39 Indeed this formula can be deduced easily by integration by parts startingfrom the identity, Im (cid:104) ( P − E ) u, u (cid:105) L ( −∞ ,x ) = 0 . Suppose now that x > c . Suppose that u is a resonant state correspond-ing to a resonance E in D h ( δ , C ) satisfying (6.6). Then the asymptoticbehaviour of the RHS of (7.1) can be given in terms of the quantity t +2 ,R ofProposition 6.1.Let χ ( x, ξ ) ∈ C ∞ ( R n ) be a cutoff function identically one in a neighborhoodof the point ( x , ξ ) ∈ Γ +2 ,R ( E ) and supported in its neighborhood. Then thefunction u − t +2 ,R χ W f +2 ,R does not have microsupport near the set { x } × R ξ ,and it follows that u = t +2 ,R χ W f +2 ,R + O ( h ∞ )locally in a neighbourhood of x = x . Hence from Proposition 5.8, one has u (cid:48) ( x ) u ( x ) − r ( x ) u ( x ) u ( x ) = | t +2 ,R | O ( h ) ,u (cid:48) ( x ) u ( x ) = | t +2 ,R | ( ih − ( ν ) (cid:48) | b +2 | + O (1)) = ih − | t +2 ,R | (1 + O ( h )) . (cid:3) Concerning the L -norm of u appearing in the previous proposition, weprove, Proposition 7.2. For any x > c and for any E ∈ [ E − δ , E + δ ] , onehas, (cid:107) u (cid:107) L ( −∞ ,x ) = I + O ( h ) , with, I := (cid:32) (cid:90) c ( E ) a ( E ) ( E − V ( x )) − / dx (cid:33) = 2 (cid:0) A (cid:48) ( E ) (cid:1) . Proof. In view of (3.3) and Proposition 5.4, our choice of normalization (6.6)for u = t ( u , u ) is made in such a way that, locally on ( −∞ , x ] (where both f − ,L and f +1 ,L contribute to u ), we have, | u | = √ πh | u − ,L | + O ( √ h ) ; u = O ( √ h ) . Moreover, we know that u is exponentially small for x < a ( E ) (both in h as h → + , and in x as x → −∞ ), and that, if we fix some x < a ( E ), then h − (cid:107) u − ,L (cid:107) L ( x ,x ) = O (1). Therefore, we have, (cid:107) u (cid:107) L ( −∞ ,x ) = πh (cid:107) u − ,L (cid:107) L ( x ,x ) + O ( √ h ) . Finally, (cid:107) u − ,L (cid:107) L ( x ,x ) can be computed by using the results of [FMW1],and one finds, (cid:107) u − ,L (cid:107) L ( x ,x ) = 4 h π (cid:90) c ( E ) − h a ( E )+ h ( E − V ( x )) − / cos (cid:18) A ( E ) + ν ( x ) h − π (cid:19) dx + O ( h ) . , A. MARTINEZ AND T. WATANABE Writing,2 cos (cid:18) A ( E ) + ν ( x ) h − π (cid:19) = 1 + sin (cid:18) A ( E ) + ν ( x )) h (cid:19) , and making an integration by parts on the second term, we obtain, (cid:107) u − ,L (cid:107) L ( x ,x ) = 2 h π (cid:90) c ( E ) − h a ( E )+ h ( E − V ( x )) − / dx + O ( h )= 2 h π (cid:90) c ( E ) a ( E ) ( E − V ( x )) − / dx + O ( h ) . 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