resonances of a two-state semiclassical schrödinger hamiltonians

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Resonances of a two-state semiclassicalSchrödinger HamiltoniansBekkai Messirdi a & Kaoutar Ghomari ba Department of Mathematics , University of Oran , Algeriab Department of Mathematics and Informatics , ENSET-Oran ,AlgeriaPublished online: 22 Feb 2007.

To cite this article: Bekkai Messirdi & Kaoutar Ghomari (2007) Resonances of a two-statesemiclassical Schrödinger Hamiltonians, Applicable Analysis: An International Journal, 86:2,187-204, DOI: 10.1080/00036810601113871

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Applicable AnalysisVol. 86, No. 2, February 2007, 187–204

Resonances of a two-state semiclassical

Schrodinger Hamiltonians

BEKKAI MESSIRDI*y and KAOUTAR GHOMARIz

yDepartment of Mathematics, University of Oran, AlgeriazDepartment of Mathematics and Informatics, ENSET-Oran, Algeria

Communicated by R. P. Gilbert

(Received 18 March 2006; in final form 9 November 2006)

The goal of this article is to generalize some travels made in (Helffer, B. and Sjostrand, J., 1986,Resonances en limite semi-classique. Memoires de la Societe Mathematique de France Ser. 2;Hunziker, W., 1986, Distorsion analyticity, and molecular resonance curves. Ann. I. H. P. (sec-tion Physique Theorique), 45, 339–358.; Messirdi, B., 1994, Asymptotique de Born-Oppenheimerpour la predissociation moleculaire (cas de potentiels reguliers). Annales de l’IHP, SectionPhysique Theorique, 61(3), 255–292.; Messirdi, B., 1993, Asymptotique de Born-Oppenheimerpour la predissociation moleculaire. These de Doctorat de l’Universite de Paris 13) to multi-dimensional two-state perturbed and semiclassical systems. More precisely, we study theSchrodinger Hamiltonians PðhÞ ¼ �h2�þ VðxÞ þ hRðx, hDxÞ on L2ðRnÞ � L2ðRnÞ, where V isflat at infinity, R is a bounded symmetric differential operator of first order and h tends to 0þ,in the case where resonances appear. Using a microlocal estimates on resolvents of P(h) andthe so-called Grushin problem the spectral study of the distorded operator of P(h) is reducedto the resolution of an analytic algebraic equation. Under the assumptions that V(x) admits arelatively compact well U and the Dirichlet realization of P(h) near U has a simple eigenvalue,it is then showed that P(h) has a unique resonance. We obtain also that the width of thisresonance is exponentially small as h tends to zero.

Keywords: Distorsion; Resonances; Microlocale estimates; Grushin problem; Resonant states

Classifications AMS: 35P15; 35Q20; 35P99; 35S99

1. Introduction

In the past few years, a lot of work has been done by several authors to study the semi-classical resonances of a scalar Schrodinger operator in the regular case (see e.g.[1,4,5,7–9]).

In particular, it is obtained in various situations the existence of resonances and theirlocalization in the complex planeby virtue of the microlocal estimations type. Someresults are also established about the exponential decrease of the widths of resonances.

*Corresponding author. Email: [email protected], [email protected]

Applicable Analysis ISSN 0003-6811 print/ISSN 1563-504X online � 2007 Taylor & Francis

http://www.tandf.co.uk/journals

DOI: 10.1080/00036810601113871

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Recently it has been proved in the singular case (see [7,10]) that the spectral study canbe reduced to that of a semiclassical pseudo-differential matrix operator depending onthe spectral parameter �, in the sense that � is a resonance if and only if 0 belongs to thespectrum of this matrix.

We investigate the existence of the resonances of the semiclassical SchrodingerHamiltonians

PðhÞ ¼ �h2�þ VðxÞ þ hRðx, hDxÞ

on L2ðRnÞ � L2ðR

nÞ, where V is a 2� 2 diagonal matrix of multiplication operators flat

at the infinity and Rðx, hDxÞ is a symmetric differential operator of first order, it repre-sents an exterior perturbation of the system and V(x) is the sum of all interactions, thegeometric form of V(x) presents a potential well in an isle [4]. We fix an energy level E0,under some assumptions we prove that P(h) has a unique resonance �(h) in the rectangle½�ðhÞ � aðhÞ, �ðhÞ þ aðhÞ� þ i½�bðhÞ, bðhÞ�, where �ðhÞ is an eigenvalue of the operatorP0ðhÞ which is the self adjoint restriction of P(h) on a small compact neighbourhoodof the well.

Our approach is similar in spirit to the one usually adopted for studying theresonances for the one-dimensional Schrodinger operator in [4] without usingcomplicated tools such as Fourier integral operators calculus with complex phase, themicrolocal estimates play here a fundamental role.

The adopted strategy is then here in a first time to get some Agmon type estimates, onthe operator P(h) and its distorted P�ðhÞ, P�ðhÞ is accessible by an appropriate vectorfield. After reduction of these operators via the Grushin problem one shows theexistence of a unique resonance of P(h) close to E0: The Agmon estimates lead then toan exponential decrease of the width of this resonance and the resonant states associated.

The article is organized as follows: in section 2, we give all possible hypotheses and weintroduce the notion of the analytic distortion and one of the resonances. In section 3,we establish some microlocal exponential a priori estimates for the operators P(h) andP�ðhÞ and their resolvants: In section 4, we study the so-called Grushin problem. Thelocalized Grushin problem near the potential well is invertible, the study of P�ðhÞ isreduced to that of a semiclassical scalar operator E�þ

� ðzÞ, this reduction is exact inthe sense that one has the following equivalence:

�2 �ðP�ðhÞÞ , E�þ� ðzÞ ¼ 0

(Here � stands for the spectrum). By the analycity of E�þ� ðzÞ, we prove the existence and

the unicity of a resonance near the fixed level E0, namely we also obtain the exponentialdecay of the width of this resonance. In section 5, we estimate the associated resonantstates of PðhÞ:

2. Hypotheses and results

We are interested in the operator:

PðhÞ ¼ �h2�þ VðxÞ þ hRðx, hDxÞ

188 B. Messirdi and K. Ghomari

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on H0 ¼ L2ðRÞ � L2ðRnÞ, when h tends to 0þ: V is a 2� 2 regular diagonal matrix

operator of multiplication on H0, such that

VðxÞ ¼V1ðxÞ 0

0 V2ðxÞ

� �

and Rðx, hDxÞ ¼P

�j j�1 C�ðxÞðhDxÞ� is a symmetric differential operator of first order

defined on H0:Let E0 > 0, €O a connected open subset of R

n and U a connected and relativelycompact open subset of €O, such that

V2ðxÞ � E0 if x2U

V2ðxÞ > E0 if x2 €OnU

V1ðxÞ ¼ E0 if x2 @ €O

V1ðxÞ > E0 if x2 €O

8>>><>>>:U is called the potential well in the ‘‘isle’’ €O:

We can define the resonances of P(h) by analytic dilation as in [1,8,9]. In this article,we instead use an equivalent definition of resonances using an analytic distortion intro-duced by Hunziker in [5] (see also [9,10]). More precisely let !2C1ðR

n,RnÞ be a

smooth vector field of Rn satisfying !¼ 0 near the potential well U and !ðxÞ ¼ x forxj j large enough. For � real small enough, the analytic distortion U� associated tothis vector field is defined by:

U�’ðxÞ ¼ ’ xþ �!ðxÞÞ J�ðxÞ�� 1=2, ’2C1

0 ðRn

� �where J�ðxÞ ¼ detð1þ �D!ðxÞÞ is the Jacobian of the transformation x ! xþ �!ðxÞ:Set

U� ¼U� 0

0 U�

� �,

if the family P�ðhÞ ¼ U�PðhÞU�1� can be extended to small enough complex values of �

as an analytic family of type ‘‘A’’ ([6,11]), we say that the operator P(h) is!-distortable. A complex number � is called a resonance of P(h) if Re � >infð�essðPðhÞÞÞ and there exists � small enough Im� > 0ð Þ, such that �2 �discP�ðhÞ:(�ess, �disc denotes respectively the essential and the discrete spectrum). For somephysical considerations, we do not consider here the resonances of P(h) which are farfrom the real axis.

On the other hand, we assume that:

(H1)

(i) V1,V2 2C1ðRn,RÞ, and V1,V2 admits a holomorphic extension in the complex strip

D� ¼ z2Cn; Re z2

Xet Im zj j < �ð1þ Re zj jÞ

n o

Resonances of two-state semiclassical Schrodinger Hamiltonians 189

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for some �>0 and whereP

is a neighbourhood of Rnn €O such thatP

¼ fx2Rn; distðx,Rn

n €OÞ < "g, " is fixed small enough and dist is the Euclidiandistance on R

n:(ii) limz2D�, zj j!þ1 VjðzÞ ¼ �j 2R and �1 < E0 < �2 it gives in particular

�essðPðhÞÞ ¼ ½�1, þ1½

(iii)P

�j j�1 C�ðxÞ�� ¼ Oð1Þ uniformly on R

n

To avoid resonances coming far from the well one uses the Virial’s hypothesis on V1

at the energy E0:

(H2)

9�0 > 0, supx2Pð2V1ðxÞ þ xrV1ðxÞÞ � 2E0 � �0

Let K0 be a compact set in Rn, U��K0 � €O, with a smooth boundary surface @K0:

P0ðhÞ denote the Dirichlet self adjoint realization of P(h) defined on L2ðK0Þ � L2ðK0Þ,with domain ðH2ðK0Þ \H1

0ðK0ÞÞ � ðH2ðK0Þ \H10ðK0ÞÞ. In particular P0ðhÞ has a real

discret spectrum.

(H3) There exist a simple eigenvalue �(h) of P0ðhÞ such that:

(i) limh!0þ �ðhÞ ¼ E0

(ii) 9aðhÞ > 0, limh!0þ aðhÞ ¼ 0, and a(h) is not exponentially small (8" > 0, 9C" > 0such that aðhÞ � ð1=C"Þe

�"=h, h>0 small enough) such that

�ðP0ðhÞÞ \ �ðhÞ � 2aðhÞ, �ðhÞ þ 2aðhÞ� ½ ¼ f�ðhÞg

(H4) U�Rðx, hDxÞU�1� can be extended to small enough complex values of � as an

analytic family of compact operators on H0.

Remark 2.1 The hypothesis (H3) is independent of the choice of the compact K0 suchthat U��K0 � €O .

Our main result is the following:

THEOREM 2.2 Under the assumptions (H1), (H2), (H3) and (H4) and for any bðhÞ � aðhÞwhich satisfies limh!0þ bðhÞ ¼ 0, there exists a unique resonance �ðhÞ of P(h) in therectangle CðhÞ ¼ ½�ðhÞ � aðhÞ, �ðhÞ þ aðhÞ� þ i½�bðhÞ, bðhÞ� such that

Im �ðhÞ�� ¼ O e�S=2h

� �where S is a geometrical constant which depends only on the potential V.

3. Microlocal estimates

In order to study the spectrum of P�ðhÞ we need to give some Agmon type estimates ondifferent resolvents of PðhÞ: This technique permits in various situations to obtain theexistence of resonances and estimate their widths. Our approach is very similar to theone usually adopted for studying interactions between two potential wells (see [3,4]).


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Let �(h) be the rectangle @CðhÞ and we denote H j, H j �ð Þ and Hj0 respectively the

Sobolev spaces HjðRnÞ �HjðR

nÞ, Hjð�Þ �Hjð�Þ and HjðK0Þ �HjðK0Þ, j ¼ 0, 1, 2,

where � is a subset of Rn:

PROPOSITION 3.1 Let be a positive Lipschitzian function on K0 such thatðr Þ2ðxÞ � ðminðV1ðxÞ,V2ðxÞÞ � E0Þþ a.e. on K0: Then, 8z2 �ðhÞ, 8" > 0, 9C" > 0,such that 8u2H0

0:

e =hðP0ðhÞ � zÞ�1u��

H10� C"e

"=h e =hu��

H00

Proof Let v ¼ ðP0ðhÞ � zÞ�1u and "ðxÞ ¼ ð1� "Þ ðxÞ for ">0 small enough, then:

Re e "=hu, e "=hv

H00¼ hrðe "=hvÞ�� 2

H00þRe ðVðxÞ � ðr "ðxÞÞ

2� zÞðe "=hvÞ, e "=hv

H0

0

þO h hrðe "=hvÞ�� 2

H00þ h e "=hv

�� 2H0

0

� �Let us denote,

�þ" ¼ fx2K0; minðV1ðxÞ,V2ðxÞÞ � E0 � "g

��" ¼ fx2K0; minðV1ðxÞ,V2ðxÞÞ � E0 < "g

Since 5 ¼ 0 on U and is lipschizian function, there exists � � 0, such that: ðxÞ � �ð Þ j��

"¼ Oð

ffiffiffi"

pÞ:

From the regularity of Vj, it follows that

Re ðVðxÞ � ðr "ðxÞÞ2� zÞðe "=hvÞ, e "=hv

H0ð��

" Þ

�� C1e2C1

ffiffi"

p=h vk k2

H00

C1 > 0: On the other hand, we have on �þ"

½VjðxÞ � ðr "ðxÞÞ2� E0� � ðVjðxÞ � E0Þ � ð1� "Þ2ðminðV1ðxÞ,V2ðxÞÞ � E0Þþ

� ½1� ð1� "Þ2�ðminðV1ðxÞ,V2ðxÞÞ � E0Þþ � "2

Thus, for z� E0j j � "2=2 and by using vk kH00� aðhÞ�1 uk kH0

0, we obtain

e "=hu��

H00e "=hv��

H00�

h2

2r e "=hv� �� 2

H00þ"2

4e "=hv�� 2

H0ð�þ" Þ

� C0"e

3C 0"

ffiffi"

p=h uk k2

H00; C 0

" > 0

and since uk kH00� ke "=hukH0

0, there exists C" > 0 such that

e "=hv�� 2

H10� C"e

3C"ffiffi"

p=h e "=hu�� 2

H0

By choosing " small enough, establishes the result. g


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Let us now consider a matrix function eW ¼ ð0 00 W2

Þ, where W2 2C10 ðfdðx,UÞ < "g, RÞ

(" is chosen rather small such that fx2 €O; dðx,UÞ < 6"g�K0) satisfyingV2ðxÞ þW2ðxÞ � E0 þ �0 on K0 thus (Vþ eW � E0 þ �0 on K0)) and denoting by eP0ðhÞthe Dirichlet self adjoint realization of P0ðhÞ þ eW on K0: If z� E0j j � �0=2 then

z2 �ðP^

0ðhÞÞ the resolvent set of P^

0ðhÞ ¼ �h2�þ Vþ eW, eP0ðhÞ � z ¼ ðP^

0ðhÞ � zÞ þ

hRðx, hDxÞ is also invertible and its inverse is bounded for h small enough because

hRðx, hDxÞ P^

0ðhÞ � z

� ��1��

��LðH0

0Þ

¼ OðhÞ

ðeP0ðhÞ � zÞ�1 is given by the convergent Neumann series

eP0ðhÞ � z� ��1

¼ P^

0ðhÞ � z

� ��1 Xþ1

n¼0

�hRðx, hDxÞ P^

0ðhÞ � z

� ��1" #n !

Therefore, we have proved the first part of the following proposition.

PROPOSITION 3.2 8x0, y0 2 0K0, 8" > 0, their exist Vx0 ,Vy0 respectively smallneighborhoods of x0 and y0, and C" > 0, such that 8z2 �ðhÞ, 8u2C1

0 ðVy0 ,C2Þ, we have

the microlocal estimate:

ðP0ðhÞ � zÞ�1u��

H2ðVx0Þ� C"e

�ðdðx0,y0Þ�"Þ=h uk kH0ðVy0Þ

in this case we say that the kernel distribution KðP0ðhÞ�zÞ�1 ðx0, y0Þ of ðP0ðhÞ � zÞ�1 is

Oðe�dðx0,y0Þ=hÞ:

Proof By applying Proposition 3.1 with eP0ðhÞ instead of P0ðhÞ and ðxÞ ¼ dðx, y0Þwhere d, is the Agmon distance associated with the degenerate metricðminðV1ðxÞ,V2ðxÞÞ � E0Þþdx

2 we have

edðx0,y0Þ=h eP0ðhÞ � z� ��1

u

�� H1ðVx0

Þ

� C"e�ð"Þ=h uk kH0ðVy0

Þ, C" > 0 ð1Þ

where lim"!0 �ð"Þ ¼ 0 and

Vx0 ¼ x2K0; dðx, x0Þ � �"�

�K0

0; Vy0 ¼ x2K0; dðx, y0Þ � "�

�K0

0

�, >0 and � ¼ if x0 ¼ y0.Moreover, �ðeP0ðhÞ � zÞ�1u ¼ ð�1=h2Þuþ 1=h2½ðVðxÞ þ eWðxÞ � zÞ þ hRðx, hDxÞ� �

ðeP0ðhÞ � zÞ�1u , but kðeP0ðhÞ � zÞ�1kLðH0,H1Þ ¼ Oðh�2Þ and by using (1) we have

�xeP0 hð Þ � z� ��1

u

�� H0ðVx0

Þ

� h�2 uk kH0ðVx0Þ

þ C"ee� "ð Þ=he�dðx0,y0Þ=h uk kH0ðVy0

Þ þC"e"=h uk kH0ðVx0

Þ

where lim"!0e�ð"Þ ¼ 0:


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If x0 6¼ y0, Vx0 and Vy0 are choosen separated, thus uk kH0ðVx0Þ ¼ 0, and

� eP0ðhÞ � z� ��1

u

�� H0ðVx0

Þ

� C"ee�ð"Þ=he�dðx0,y0Þ=h uk kH0ðVy0

Þ

if x0 ¼ y0, Vx0 ¼ Vy0 and for h>0 small enough

h�2 uk kH0ðVx0Þ � C"e

e�ð"Þ=he�dðx0,y0Þ=h uk kH0ðVy0Þ

Finally, we obtain

� eP0ðhÞ � z� ��1

u

�� H0ðVx0

Þ

� 2C"ee�ð"Þ=he�dðx0,y0Þ=h uk kH0ðVy0

Þ ð2Þ

By using (1) and (2), we can finally deduce the microlocal estimate of ðeP0ðhÞ � zÞ�1:

eP0ðhÞ � z� ��1

u

�� H2ðVx0

Þ

� 2C"ee�ð"Þ=he�dðx0,y0Þ=h uk kH0ðVy0

Þ ð3Þ

Now the main idea of our proof consists in comparing ðP0ðhÞ � zÞ�1 with theresolvent ðeP0ðhÞ � zÞ�1 acting on H2

0. For this purpose we introduce two additionalfunctions {1,{2 2C1

0 ðK0Þ such that

Supp{1 �fx2K0; dðx,UÞ < 3"g; Supp{2 �fx2K0; dðx,UÞ < 5"g

{1 ¼ 1 onfx2K0; dðx,UÞ � 2"g; {2 ¼ 1 onfx2K0; dðx,UÞ � 4"g

�

So, 8z2 �ðhÞ

ðP0ðhÞ � zÞ�1¼ ð1� {1Þ eP0ðhÞ � z

� ��1

ð1� {2Þ þ ðP0ðhÞ � zÞ�1{2

þ ðP0ðhÞ � zÞ�1{2½P0ðhÞ,{1� eP0ðhÞ � z

� ��1

ð1� {2Þ

Using now the Proposition 3.1 with ðxÞ ¼ dðx,UÞ, then 8u2C10 ðVy0 ,C

2Þ

edðx,UÞ=hðP0ðhÞ � zÞ�1{2u

�� H1ðVx0

Þ� C"e

"=h edðx,UÞ=hu��

H0ðVy0Þ

dðx,UÞ ¼ dðx0,UÞ þ Oð"Þ on Vx0 and dðx,UÞ ¼ Oð"Þ on Vy0 \Supp{2, thus

ðP0ðhÞ � zÞ�1{2u

�� H1ðVx0

Þ� eC"eðe�ð"Þ�d0ðx0, y0ÞÞ=h uk kH0ðVy0

Þ

eC" > 0. We obtain the same semiclassical estimate for �ðP0ðhÞ � zÞ�1{2, and thus

KðP0ðhÞ�zÞ�1

{2¼ O e�dðx0,UÞ=h

� �


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By the same manner, we have Kð1�{1ÞðP0ðhÞ�zÞ�1

ð1�{2Þðx0, y0Þ ¼ Oðe�dðx0,y0Þ=hÞ:

In the other hand,

ðP0ðhÞ � zÞ�1{2½P0ðhÞ,{1� eP0ðhÞ � z

� ��1

ð1� {2Þu

�� H2ðVx0

Þ

� C"e�ðdðx0,UÞ�"Þ=h ½P0ðhÞ,{1� eP0ðhÞ � z

� ��1

ð1� {2Þu

�� H2ðVx0

Þ

Since ½P0ðhÞ,{1� is localized on Supp 5{1ð Þ near the well U, then

KðP0ðhÞ�zÞ�1

{2½P0ðhÞ,{1�ðeP0ðhÞ�zÞ�1ð1�{2Þ

¼ O e�ðdðx0,UÞþdðy0,UÞÞ=h� �

thus,

ðP0ðhÞ � zÞ�1{2u

�� H2ðVx0

Þ

� C�

"e~�ð"Þ=h e�dðx0,y0Þ=h þ e�dðx0,UÞ=h1Supp {2ð Þ þ e�ðdðx0,UÞþdðy0,UÞÞ=h

� �uk kH0ðVy0

Þ

� C"e�ð"Þ=he�dðx0,y0Þ=h uk kH0ðVy0

Þ g

We consider now the elliptic operator

eP hð Þ ¼ P hð Þ þ eW xð Þ ¼ bP hð Þ þ hRðx, hDxÞ

where bP hð Þ ¼ diagðbP1 hð Þ, bP2 hð ÞÞ, bP1 hð Þ ¼ �h2�þ V1 and bP2 hð Þ ¼ �h2�þ V2 þW2.Consider also the distorted operator eP�ðhÞ ¼ U�ePðhÞU�1

� of ePðhÞ definite by a C1

vector fields ! such as Supp!ð Þ �RnnK0 and !ðxÞ ¼ x in R

nnK1 where U��

K0 � 0K1 � K1 �� €O:

Thus, eP�ðhÞ ¼ diagðbP1,� hð Þ, bP2,� hð ÞÞ þ hR�ðx, hDxÞ where bP1,� hð Þ ¼ U�bPj hð ÞU�1

� ,j¼ 1, 2 and

R�ðx, hDxÞ ¼ U�Rðx, hDxÞU�1� ¼ L�ðx, hDxÞ þ S�ðx, hDxÞ

L�ðx, hDxÞ ¼X�j j�1

1

ð1þ �Þ �j jC� 1þ �ð Þxð Þh �j j@�x

S�ðx, hDxÞ ¼X�j j�1

b�ðx,�Þh�j j@�x þ

X�j j¼1

hd�ðx,�Þ

L� and S� are differential operators of first order whose coefficients are C10 ðK

0

2,C2)

(where K2 is a compact set of Rn) analytic for �j j small enough satisfies

82Nn; sup

x2Rnb�ðx,�Þ�� ¼ Oð1Þ, sup

x2Rnd�ðx,�Þ�� ¼ Oð �j jÞ


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In particular,

U�Rðx, hDxÞU�1�

bP� hð Þ � z� ��1

�� L H0ð Þ

¼ O 1ð Þ

PROPOSITION 3.3 For any complex z sufficiently near E0 and �2C, Im� 6¼ 0 suchthat �j j small enough, the operator ðeP� hð Þ � zÞ is invertible with bounded inverse fromH2 to H.

Proof It is sufficient to prove that ðbP� hð Þ � zÞ is invertible with bounded inverse fromH2 to H0.

Let 1 2C10 ð €OÞ and 2 2C1

0 ðRnÞ such that

1 xð Þ ¼ 1 on K1; Suppð2Þ �Rnn K1

x2Rn; 1 xð Þ ¼ 1

� [ x2R

n; 2 xð Þ ¼ 1�

¼ Rn

�

then ðV1 xð Þ � E0ÞjSupp 1ð Þ� �0 > 0:

Let us get for u ¼ ðu1, u2Þ and i2 1, 2f g, Ai1 ¼ h1ðbPi hð Þ � E0Þui,1uiiL2 R

nð Þ, we have

ðbPi hð Þ � E0Þ1ui,1ui

D EL2 R

nð Þ

�� h2 r 1uið Þ�� 2

L2 Rnð Þþ �0 1uik k2L2 R

nð Þ

and

1, bPi hð Þ

h iui,1ui

D EL2 R

nð Þ

�� ¼ O ðh3=2 ruik kL2 Rnð Þ þ h1=2 uik kL2 R

nð ÞÞ2

� �Consequently,

Ai1

�� h2 r 1uið Þ�� 2

L2 Rnð Þþ �0 1uik k2L2 R

nð Þ

� C2 h3=2 ruik kL2 Rnð Þ þ h1=2 uik kL2 R

nð Þ

� �2C2 > 0: If Ai

1,� ¼ h1ðbPi,� hð Þ � E0Þui,1uiiL2 Rnð Þ, then

Ai1,� ¼ Ai

1 þ �j jX

�j j�2, j j�2

1eai�ðx,�Þh �j j@xui,1ui

D EL2 R

nð Þ

where the coefficients eai� are analytic and bounded with respect to x and �:However,

X�j j�2, j j�2

1eai�ðx,�Þh �j j@xui,1ui

D EL2 R

nð Þ

��

¼ O h h r 1uið Þ�� 2

L2 Rnð Þþ 1uik k2L2 R

nð Þ þ h2 ruik k2L2 Rnð Þ þ uik k2L2 R

nð Þ

� ��


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Thus,

Ai1,�

�� 021uik k2L2 R

nð Þ þh2

2r 1uið Þ�� 2

L2 Rnð Þ

� C3 h3=2 ruik kL2 Rnð Þ þ h1=2 uik kL2 R

nð Þ

� �2C3 > 0: Let us note too Ai

2,� ¼ h2ðbPi,� hð Þ � E0Þui,2uiiL2 Rnð Þ, then as previously

2, bPi,� hð Þ

h iui,2ui

D EL2 R

nð Þ

�� ¼ O h3=2 ruik kL2 Rnð Þ þ h1=2 uik kL2 R

nð Þ

� �2� �

We have ! xð Þ ¼ x and W xð Þ ¼ 0 on supp(2Þ, thus

bPi,� hð Þ ¼ �h2

ð1þ �Þ2�þ Viðð1þ �ÞxÞ

and

Re 1þ �ð Þ2ðbPi,� hð Þ � E0Þ2ui,2ui

D EL2ðRnÞ

� h2 r 2uið Þ�� 2

L2ðRnÞ�C4 2uik k2L2ðRnÞ

C4 > 0: On the other hand

Im 1þ �ð Þ2ðbPi,� hð Þ � E0Þ2ui,2ui

D EL2ðRnÞ

¼ Im 1þ �ð Þ2ðViðð1þ �ÞxÞ � E0Þ2ui,2ui

L2ðRnÞ

By using the Virial’s assumption (H2) on the potential V1 and the development inTaylor series of fð�Þ ¼ 1þ �ð Þ

2 V1ðð1þ �ÞxÞ � E0½ � we obtain, for |�| small enough,on Supp(2Þ

Im 1þ �ð Þ2ðV1ðð1þ �ÞxÞ � E0Þ2u,2u

L2ðRnÞ

�� 0 Im�j j 2u1k k2L2ðRnÞ

On the other hand ðV2ðð1þ �ÞxÞ � V2ðxÞÞ ¼ Oð �j jÞ, and

Im 1þ �ð Þ2ðV2ðð1þ �ÞxÞ � E0Þ2u2,2u2

L2ðRnÞ

�� Im 1þ �ð Þ

2�� 0

22u2k k2L2ðRnÞ� Im�j j2

�022u2k k2L2ðRnÞ


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Thus

1þ �ð Þ2 bPi,� hð Þ � E0

� �2ui,2ui

D EL2ðRnÞ

��

1

C5Im�j j2 h2 r 2uið Þ

�� 2L2ðRnÞ

þ 2uik k2L2ðRnÞ

� �2C5 > 0: We finally obtain

Ai2,�

�� 1

C6Im�j j h2 r 2uið Þ

�� 2L2ðRnÞ

þ 2uik k2L2ðRnÞ

h i� C6 h3=2 ruik k2L2ðRnÞ þ h1=2 uik k2L2ðRnÞ

h iC6 > 0: Thus

Ai1,�

�� þ Ai2,�

�� 1

C7Im�j j h2 ruik k2L2ðRnÞ þ uik k2L2ðRnÞ

� �C7 > 0: Since jui

�� L2ðRnÞ

� uik kL2ðRnÞ, we obtain if z� E0j j Im�j j,

bPi,� hð Þ � z� �

ui

�� L2ðRnÞ

�Im�j j

C8uik kL2ðRnÞ, C8 > 0

which prove that ðbP� hð Þ � zÞ is invertible for z sufficiently near to E0, and

bP� hð Þ � z� ��1��

LðH0,H2Þ

�C9

Im�j j

C9 > 0: For h Im�j j, we have also kðbP� hð Þ � zÞ�1kLðH0,H2Þ ¼ Oðh�1Þ: g

COROLLARY 3.4 For � complex small enough with Im� 6¼ 0, z complex sufficiently near

to E0 and x0, y0 2K0

0 we can by the same argument establish that

KðeP�ðhÞ�zÞ�1

ðx0, y0Þ ¼ Oðe�edðx0, y0Þ=hÞ

where edðx0, y0Þ ¼ minðdðx0,y0Þ, dðx0, @K0Þ þ dð y0, @K0ÞÞ:

4. The Grushin problem

The Grushin problem is a well-known method of reduction used in [2] and that wepresent here in a slightly different way.


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For z� E0j j small enough, we consider the matrix operator

P0ðzÞ ¼P0ðhÞ � z R�

0

Rþ0 0

� �

defined on ðH2ðK0Þ \H10ðK0ÞÞ

2� C, with

R�0 u

� ¼ u�’0, 8u� 2C

Rþ0 u ¼ u, ’0

H0, 8u2 ðH2ðK0Þ \H1

0ðK0ÞÞ2

(

Where ’0 is a normalised eigenfunction associated to the energy level �(h).It is thus easy to check that P0ðzÞ is invertible for z2CðhÞ of inverse denoted by

P�10 ðzÞ ¼

E0ðzÞ Eþ0 ðzÞ

E�0 ðzÞ E�þ

0 ðzÞ

!

such that for ðv, vþÞ 2 ðL2ðK0ÞÞ2� C, we have

E0ðzÞv ¼ P00ðhÞ � zÞ�1

ðv� v, ’0

H0’0

� �Eþ0 ðzÞv

þ ¼ vþ’0 ¼ R�0 v

þ

E�0 ðzÞv ¼ v, ’0

H0¼ Rþ

0 v

E�þ0 ðzÞvþ ¼ ðz� �ðhÞÞvþ

8>>>>><>>>>>:where P0

0ðhÞ is the restriction of P0ðhÞ on the orthogonal of ’0: In particular, we get thereduction result

z2 �ðP0ðhÞÞ \ ½�ðhÞ � aðhÞ, �ðhÞ þ aðhÞ� , E�þ0 ðzÞ ¼ 0

Now, the main idea of our proof consists in estimating the resolvent P0ðhÞ � zð Þ�1: For

this purpose we investigate the exponentially small estimates on the terms of the matrixP�1

0 ðzÞ: We have the following result

PROPOSITION 4.1

(i) KE0ðzÞðx, yÞ ¼ Oðe�dðx, yÞ=hÞ

(ii) KEþ

0ðzÞðxÞ ¼ Oðe�dðx,UÞ=hÞ

(iii) KE�0ðzÞðyÞ ¼ Oðe�dðy,UÞ=hÞ

uniformly with respect to h small enough and z nearly to E0.

Proof Let e�ðhÞ be the rectangle @ð½�ðhÞ � ð3aðhÞ=2Þ, �ðhÞ þ ð3aðhÞ=2Þ� þ i½�bðhÞ, bðhÞ�Þ.Then, 8z2 �ðhÞ

E0ðzÞ ¼ �1

2�i

Ie�ðhÞðt� zÞ�1

ðt� P0ðhÞÞ�1dt


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Using Proposition 3.2 we have KðP0ðhÞ�tÞ�1ðx, yÞ ¼ Oðe�dðx, yÞ=hÞ and by the fact that

8t, z2e�ðhÞ, 1=ðt� zÞ�� 2=aðhÞ � 2C"e

"=h, we obtain (i).(ii) Let ðxÞ ¼ ð1� "Þdðx,UÞ, then by using the Proposition 3.2 we have

Re e =hðP0ðhÞ � �ðhÞÞ’0, e =h’0

H00

��

h2

25 e =h’0� �� 2

H00þ"2

4e =h’0�� 2

Hð�þ" Þ�C0

"e3C0

"ðffiffi"

p=hÞ

Moreover,

e =h’0�� 2

Hð��" Þ� C10e

2C10ðffiffi"

p=hÞ

C10 > 0. then

h2

25 e =h’0� �� 2

H00þ"2

4e =h’0�� 2

H00� C11e

C12ðffiffi"

p=hÞ

C11 > 0. and

e =h’0��

H10¼ O e�ð"Þ=h

� �with lim"!0 �ð"Þ ¼ 0. We deduce that kedðx,UÞ=h’0kH1

0¼ Oðe"=hÞ.

Let e 2C1ðK0Þ such that ke � kL1ðK0Þ� ": By using P0 hð Þ’0 ¼ � hð Þ’0 and the

fact that

h e~ xð Þ=hRðx, hDxÞ’0

�� H0

0

¼ eO e~ xð Þ=h’0

�� H0

0

þ e~ xð Þ=hr’0

�� H0

0

� �

we obtain

�ðe~ xð Þ=h’0Þ

�� H0

0

¼ eO e~ xð Þ=h’0

�� H0

0

þ rðe~ xð Þ=h’0Þ

�� H0

0

� �¼ eO 1ð Þ

and

e~ xð Þ=h’0

�� H2

0

¼ eO 1ð Þ

where eO 1ð Þ ¼ Oðee�ð"Þ=hÞ with lim"!0 e�ð"Þ ¼ 0: g


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For z� E0j j and Im�j j small enough such that Im� 6¼ 0, we consider the distortedGrushin operator

P�ðzÞ ¼P�ðhÞ � z R�

0

Rþ0 0

� �

with 2C10 ðK0Þ and ¼ 1 on fx2K0; dðx, @K0Þ � "g, ">0 is small enough: It is then

easy to see that P�ðzÞ is invertible fromH2ðRnÞ �H2ðR

nÞ � C into L2ðR

nÞ � L2ðR

nÞ � C

for any z2CðhÞ:Let us pose

F�ðzÞ ¼E0ðzÞ þ eP�ðhÞ � z

� ��1

ð1� Þ Eþ0 ðzÞ

E�0 ðzÞ E�þ

0 ðzÞ

0@ 1A 2C1

0 ðK0Þ, such that Supp �fx2K0; dðx,UÞ � ðS0=2Þg and ¼ 1 on fx2K0;dðx,UÞ � ðS0 � "Þ=2g, where S0 ¼ dðU, @K0Þ > 0: In particular, !¼ 0 onSuppW \ Supp ð Þ and ¼ :

PROPOSITION 4.2

P�ðzÞF�ðzÞ ¼ 1þK�ðzÞ

with

K�ðzÞ ¼eOðe�S0=2hÞ eOðe�S0=hÞeOðe�S0=2hÞ eOðe�2S0=hÞ

!

uniformly for h> 0 small enough.

Proof Writing

P�ðzÞF�ðzÞ ¼A B

C D

� �,

we have

A ¼ ðP�ðhÞ � zÞ½E0ðzÞ þ ðeP�ðhÞ � zÞ�1ð1� Þ� þ R�

0 E�0 ðzÞ

B ¼ ðP�ðhÞ � zÞEþ0 ðzÞ þ R

�0 E

�þ0 ðzÞ

C ¼ Rþ0 E0ðzÞ þ Rþ

0 ðeP�ðhÞ � zÞ�1

ð1� ÞD ¼ Rþ

0 Eþ0 ðzÞ

R�0 E

�0 ¼ h:, ’0iH0’0 and ½P�ðhÞ,� ¼ PðhÞ,½ � because ½P�ðhÞ,� is supported on

Supp5ð Þ:Moreover, since ðP�ðhÞ � zÞ ¼ ðeP�ðhÞ � zÞ � eW, we have

A ¼ 1þ ½PðhÞ,�E0ðzÞ � eWðeP�ðhÞ � zÞ�1ð1� Þ


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KE0ðzÞðx, yÞ ¼ Oðe�dðx, yÞ=hÞ, x2 Supp ½PðhÞ,�ð Þ is localized near @K0, y2 Supp islocalized near the ball BdðU, ðS0=2ÞÞ ¼ x2K0; dðx,UÞ � ðS0=2Þ

� , thus

½PðhÞ,�E0ðzÞ ¼ eO e�S0=2h� �

On the other hand,

eW eP�ðhÞ � z� ��1

ð1� Þ

¼ eW 1� e � � eP�ðhÞ � z� ��1

ð1� Þ þ eP�ðhÞ � z� ��1 eP�ðhÞ,e h i eP�ðhÞ � z

� ��1

ð1� Þ

� �¼ eW eP�ðhÞ � z

� ��1 eP�ðhÞ,e h i eP�ðhÞ � z� ��1

ð1� Þ

where e 2C10 ðK0Þ with Suppðe Þ � fx2K0; dðx,UÞ � ðS0=2Þ � "g, e ¼ 1 on x2K0;f

dðx,UÞ � ðS0=2Þ � 2"g and e ¼ e :But, ðeP�ðhÞ � zÞ�1

ð1� Þ ¼ Oð1Þ, y2 Supp ð½eP�ðhÞ, e �Þ is localized nearly to y2K0;dðy,UÞ ¼ ðS0=2Þg: x2 Suppð eWÞ is localized near the well U, and K

ðeP�ðhÞ�zÞ�1ðx, yÞ ¼

Oðe�edðx, yÞ=hÞ:

In this case, edðx, yÞ � dðx, @K0Þ � dðx, @K0Þ þ dðy, @K0Þ, then edðx, yÞ ¼ dðx, yÞ.

Consequently, A ¼ 1þ eOðe�S0=2hÞ:

By the same manner we show that B ¼ eOðe�S0=hÞ, C ¼ eOðe�S0=2hÞ

and D ¼ 1þ eOðe�2S0=hÞ: g

As a consequence of Proposition 4.2, the Neumann seriesP1

j¼0ð�K�ðzÞÞj is absolutely

convergent on LðH0Þ for h small enough and

E�ðzÞ ¼ F�ðzÞX1j¼0

ð�K�ðzÞÞj

is the right inverse of P�ðzÞ:We remark that if we denote

G�ðzÞ ¼ E0ðzÞþ ð1� Þ eP�ðhÞ � z

� ��1

Eþ0 ðzÞ

E�0 ðzÞ E�þ

0 ðzÞ

0@ 1Awe construct also a left inverse of P�ðzÞ:

P�ðzÞ is then invertible, it is inverse E�ðzÞ is now denoted by

P�1� ðzÞ ¼ E�ðzÞ ¼

E�ðzÞ Eþ� ðzÞ

E�� ðzÞ E�þ

� ðzÞ

!¼ F�ðzÞ 1þ eO e�S0=2h

� �h iIn particular, we obtain the identity

E�þ� ðzÞ ¼ E�þ

0 ðzÞ þ eO e�S0=2h� �


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and the reduction result

z2 �ðP�Þ , E�þ� ðzÞ ¼ 0

E�þ� ðzÞ ¼ ðz� �ðhÞÞ þ rðz, hÞ, where rðz, hÞ ¼ eOðe�S0=2hÞ is an analytic function on C(h)

with respect to z. By using the Cauchy theorem on the rectangle C(h), we have

@rðz, hÞ

@z

�� sup rðz, hÞ�� aðhÞ

� eC"e"=hso

@

@zE�þ� ðzÞ ¼ 1þ eO e�S0=2h

� �6¼ 0

Thus, E�þ� ðzÞ ¼ 0 if z ¼ �ðhÞ þ eOðe�S0=2hÞ for z2 ½�ðhÞ � ðaðhÞ=2Þ, �ðhÞ þ ðaðhÞ=2Þ� þ

i½�ðbðhÞ=2Þ, ðbðhÞ=2Þ� and conversely. By using the implicit function theorem theequation E�þ

� ðzÞ ¼ 0 has necessarily a unique solution �(h) such that

E�þ� ð�ðhÞÞ ¼ 0

�ðhÞ ¼ �ðhÞ þ eO e�S0=2h� �(

We deduce that P(h) admits a unique resonance in the rectangle C(h) andj�ðhÞ � �ðhÞj ¼ eOðe�S0=2hÞ.

In particular, since �(h) is real we obtain

Im �ðhÞ�� ¼ eO e�S0=2h

� �Remark 4.3 If we note S ¼ dðU, @ €OÞ K0 ¼ K0," such that @K0,"�fdðx, @ €OÞ � "g, weobtain jIm �ðhÞj ¼ eOðe�2S=hÞ:

5. The resonant states

The resonant states of P(h) are the eigenfunctions of P�ðhÞ associated with the reso-nance �(h). Let �� be the spectral projection operator associated with P�ðhÞ and �(h)

�� ¼1

2i�

I�ðhÞ

ðz� P�ðhÞÞ�1dz

Then the set of the resonant states is the range of ��, i.e., Im��: Let also 2C10 ðK0Þ

such that the field !¼ 0 on Supp and ¼ 1 on fx2K0; dðx, @K0Þ � "g: ’0 is agood approximation of the resonant state ��ð’0Þ, more precisely we have


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PROPOSITION 5.1

��ð’0Þ � ’0��

H0¼ eO e�S0=h� �

Proof We have

ðPðhÞ � �ðhÞÞð’0Þ ¼ ½PðhÞ,�’0 ¼ r0; r0j jH0¼ eO e�S0=h

� �and

1

ð�ðhÞ � zÞð’0Þ ¼ ðP�ðhÞ � zÞ�1

ð’0Þ þ1

ð�ðhÞ � zÞðP�ðhÞ � zÞ�1r0

Consequently,

��ð’0Þ ¼ ’0 þ1

2i�

I�ðhÞ

1

ð�ðhÞ � zÞðP�ðhÞ � zÞ�1r0dz

Let R�ðzÞ ¼ ðP0ðhÞ � zÞ�1þ ð1� ÞðeP�ðhÞ � zÞ�1 and 2C10 ðK0Þ, such that

Supp ð Þ � fx2K0; dðx,UÞ � ðS0=2Þg, ¼ 1 on fx2K0; dðx,UÞ � ðS0 � "Þ=2g and¼ 1 on Supp ð Þ: Then,

R�ðzÞðP�ðhÞ � zÞ ¼ þ ðP0ðhÞ � zÞ�1½,PðhÞ� þ ð1� Þ

� ð1� Þ eP�ðhÞ � z� ��1 eW ¼ 1þ eO e�S0=2h

� �Consequently, 8z2 �ðhÞ,

ðP�ðhÞ � zÞ�1¼ 1þ eO e�S0=2h

� �� R�ðzÞ

Since

ðP0ðhÞ � zÞ�1r0 ¼ eO e�3S0=2h� �

and

ð1� Þ eP�ðhÞ � z� ��1

r0eO r0k kH0

� �¼ eO e�S0=h

� �then

ðP�ðhÞ � zÞ�1r0 ¼ 1þ eO e�S0=2h� ��

R�ðzÞr0 ¼ eO e�S0=h� �

which give the result. g


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Acknowledgement

Investigations supported by University of Oran Essenia, Algeria. Research TopicsCNEPRU B3101/02/03.

References

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[3] Helffer, B. and Sjostrand, J., 1984, Multiple wells in the semi-classical limit I. Communications in PartialDifferential Equations, 9, 337–408.

[4] Helffer, B. and Sjostrand, J., 1986, Resonances en limite semi-classique. Memoires de la SocieteMathematique de France Ser. 2, 24–25, 1–228.

[5] Hunziker, W., 1986, Distorsion analyticity, and molecular resonance curves. Ann. I. H. P. (sectionPhysique Theorique), 45, 339–358.

[6] Kato, T., 1966, Perturbation Theory for Linear Operators (Berlin: Springer-Verlag).[7] Martinez, A. and Messirdi, B., 1994, Resonances of diatomic molecules in the Born-Oppenheimer

approximation. Communications in Partial Differential Equations, 19, 1139–1162.[8] Messirdi, B., 1994, Asymptotique de Born-Oppenheimer pour la predissociation moleculaire (cas de

potentiels reguliers). Annales de l’IHP, Section Physique Theorique, 61(3), 255–292.[9] Messirdi, B., 1993, Asymptotique de Born-Oppenheimer pour la predissociation moleculaire. These de

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Born-Oppenheimer approximation. Journal of Mathematical Physics, 46, 103506.[11] Reed, M. and Simon, B., 1978, Methods of Modern Mathematical Physics. T. 4 (New York: Academic

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resonances of a two-state semiclassical schrödinger hamiltonians

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