MATHEMATICAL THEORY OF QUANTUM TUNNELING

AT POSITIVE TEMPERATURE

Branislav Vasilijevic

A thesis subrnit ted in conformity with the requirements for the degree of Doctor of Phüsophy Graduate Department of Mathematics

University of Toronto

O Copytight by Branislav Vasilijevic 2001

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MATHEMATICAL THEORY OF QUANTUM TUNNELING

AT POSITrVE TEMPERATURE

P h.D. t hesis of Branislav Vasilijevic

Department of Mathematics, Cniversity of Toronto, 2001

The goal of this work is to initiate mathematical theory of tunneling

at positive temperatures. The mathemat ical framework we develop

here rests on the notion of resonance /ree eneqy, F(P) (where B is

inverse temperature). This quantity plays the role of the resonance

energy at zero temperature. We prove that the probability of es-

cape of the particle fiom a potential well due to tunneling can be

expressed as P(t ) = 1 - p( t ) where roughly:

modulo O (A2) +O ((!)QD), where A is an exponentidy small (in !)

quantity specified in Section 8. Here h is the Planck constant divided

by 2s and

r = -2Im(F(P)) ,

is the " width" of the resonance free energy.

Furthemore, we give a semi-classical bound on r:

for some C > O, where Ss is the action of the instanton of penod AP.

TABLE OF CONTENTS:

1. INTRODUCTION I

2. HAMILTONWIYS AND RESULTS 7

3. DEFORncIATION FAbiILY FOR H 24

4. RESONANCE FREE EAERGY 18

5. FEYNMAN-KAC THEOREM FOR RESON ANCES 23

6. EXPONENTIAL DECAY OF DEFOR'iiIED EIGfiWbTCTIONS 26

7. SPECTRAL ANALYSE OF THE FAiiIILY HOA 36

8. TUNNELNG PROBABIUTY XiW RESONANCE FREE ENERGY 53

9. SEMICLASSIC AL APPROXIMATION 66

10. APPENDIX: REVIEW OF RESONANCE THEORY 68

11. SUPPLEMENT: FLOW D U G W I AND NOTATION 78

REFERENCES 80

1 INTRODUCTION

The goal of this paper is to initiate mathematical theory of tunnelhg at p o s

itive temperatures. The mathematical framework we develop here rests on

the notion of free monance energy, F(p) (where P is inverse temperature),

which we introduce in this paper. This quantity plays the role of the r-

nance energy at zero temperature. We prove that the probability of escape of the psrticle from a potential well due to tunneling caa be expressed as

P(t ) = 1 - p(t) where roughly:

h W modulo O (A2) + O ((7) ) , where A is an exponentidy smaU (in i) quantity specified in Section 8 (see Theorern 8.1 below). Here h is the Planck constant divided by 2sr, considered as a srnad parameter, and

the "width" of the fiee energy. The formula above is taken for granted in

physics literature and is used in condensed matter physics ([A, CL, LOI) and cosmology ([Ll-21, [VS]) in order to andyze the tunnelhg proceas. However,

it was never justfied or analyzed systematically, not to mention rigorously.

For instance, the expression for r (see (1.5) below) or for F(P) was never

connected directly to the underlying Schroedinger operator as it is done in

our work. Eùrthermore, we give a semi-classical bound on r:

for some C > O. Here SB is the action of the instanton of period î$ (see

(2.22) and discussion preceding it).

Since the notion of temperature pertsins to equiiibrinm states of systems

with Ûthite uumber of degrees of W o m wliile the tunneling is obviously

a non-equilibrium process and a quantum particle has only tkee degrees of fkeedom we have to dari& what we mean by tmeling at a positive temper- atnre. The latter t e m refm to the process of tunneli~g of a particle which

is either in a contact with a reservoir (Le. a system of idhite degrees of freedom, e.g. photon or phonon gas) which at time t = O is in a state of equilibriurn at temperature T, or which is initiaIly thermslized.

In the case of initial thermalization the temperature is introduced through the initial condition and has no other &ect on the dynamics of the system. This corresponds to the following physicaI situation: the system is prepared by putting it in contact with a resemoir at temperature T, and at t=O t"=er-

voir is removed and the particle system is leR to evolve on its own, or, put dinerently, the effect of the rese~oir on the particle is ignored. This physical

situation is of interest in its own right as well as for the reason of giving a good approximation to the process of tunnelhg with a thermal reservoir included. Indeed, usually the coupling between the partide and the resemoir

is, on one hand, dc i en t ly strong so that the reservoir maintains the particle "inside the well" in a state of ( a p p r h a t e ) equilibrium and, on the other

hand, is suniciently weak so that it yields only a srnad perturbation to the tunneling procas. In other words, usually we have:

where Ttunding = r-' and T,tQZatim = (couphg constant)-L are the char- acteristic times of the tunneling and relaxation, respectively, and Tm is the characteristic t h e of the particle system, Say &, where A E is a mean gap between energy levels. Thus (for smaU enough ternperatures) the coupling

between the particle and the reservoir dnring the process of tumehg can be neglected in the leading approximation ([GWH]) .

Finally, we address the resonance nature of the process of tunneliog.

Naively one can thmk about turmeling at positive temperatme as follows.

InitiaiIy the pasticle is insde the well and is in each of the "well states"

$,, with the Gibbs probabilities Z-L e-flh, where En is the energy of the

state $,, and p is the inverse temperature and Z = En e-B" the normal- ization factor. Since the particle can tunnel from each state @n, these states must be replaceci by the resonance states with the resonance eigenvalues

- z ~ ~ ~ , where w En and r n d i n 9 is the "width" of the rese nance fevel. The quantity i r d i n 9 is interpreted a s a probability of decay per unit time due to the tunaeiing from the level $,,. Thus it is not surprising that the probability of tunnelhg per unit time at temperature T = is given appraximately by :

which coincides in the leadhg order semic1assicdy with our expression (1.2).

In fact, this is how physicists derive (1.1).

In this paper we study the tunneling of initially thermalized particle and derive relations (1.1)-(1.3) in this case. This allows us to use a fau'y elemen- tary analysis based on spectral theory of self-adjoint and non-self-adjoint Schroedinger operators.

Thus we consider a quantum particle in Rd moving under the hduence

of an externd potentisl V ( x ) , which is a red function on Rd. The dynamics of such s particle is describeci by the Schroedinger operator:

acting in the space L2 (Rd). Here A = y''!l qi is the Laplacian operator

on 3, we use hensionless units and consider tr (whose origin is in the Planck's constant or couphg constant) as a dimensionles small parameter. (The latter is done for simplicity as there are other n a t d parameters related to the shape of potentid barrier which do the same job as 5). We also set

mass to i. We assume that V ( x ) is sueh that H is a &-adjoint operator. Moreover

we assnme that V(x) is of the fom:

Le. it has a compact well region separated by a nnite barrier from the un- bounded domain in which the values of the potential are below the bottom of the well. (Technid restrictions on V(z) are given in Section 2.) We are

interesteci in the probability of decay of states which aie initidy locaiized in the wd.

Since we are stndying the dynamics at positive temperature, we have to consider mixed states of the particle. These are given by density matrices, p,

i.e. non-negative, traceclass operators on L2 (Rd) nomaüzed as Tr(p) = 1. The evolution of these states is given by the von Neumann equation:

subject to the initial condition p(O) = p,, where p, is an approximate Gibbs state of the w d (see Section 2 for an exact definition).

The probability that the state p(t ) which is initially localized in the weii region, W, is stiU Iocaüzed there at time t is given by:

where x is a chosen particle observable IocaIized in W. There are hm n a t d choices of X : the characteristic fiinction, x = x(W), of the well W and the projection of the weIl eigenfnnctions which are defined with a hdp of an ainriliary Hamiltonian, cded the reference Hamiltonian (see Section 2).

Then the probability of the partide escaping h m the well is given by P(t) = 1 - p(t) = Tr ( ~ p ( t ) ) , where X = 1 - X.

Next, we list some of the landmarks in the development of the subject. Landmarks:

1920s: Quantum tunneling, WKB apprmcimation, Fermi golden d e ;

1928: Gamow [Ga] describes adecay of nuclei as a quantum tunneling process;

1967: Langer [La], Coleman CO^], LevitSmolenski use path integral to study quantum t u m e h g in the contact of Statistical Mechanics (phase transitions), QFT (unst able vacua) and nuclear physics, respectively.

1971-1973: Balslev, Combes [BC] and Simon [SI] use dilation analyt- icity to study resonances of Scroedinger operators;

19'794986: Simon [S2], Sigal [Sil], Hunziker [Hu] and Cycon [Cy] ex- tend dilation analyticity framework to generd vector fields;

1981: Simon [S3] produces a rigorous treatment of tunneling;

1981-1990: Guth, Linde [Ll,L2] and others extend Langer's results to the study of cosmology;

1981-1984: AfEedc [A], Caldeira-Leggett [CL], Larkin-Ovchinnikov [LO,L02] and many others initiate investigation of tunnelixig at T > O in m a c r m tems, such as superconductors and superfiuids;

1986: CombesDuclos-Klein-Seiler [CDKS], Helffer-Sjoestrand [HeSj] and HislopSigal [HSl] prove existence of tnnneling resonances and es- timate their iifetimes;

1986-present: b ~ h g and ttl~l~leling resonmces is an active field of reseaxch in mathematics, for T = O (see e.g. PZ,CS,MS,Sw for recent papers), and in ph@= for T 2 O (see e.g. [EMS,ES,KL]).

QTPT 6

The text is orgBriized as follows. In Section 2 we describe the dass of Schroedinger operators under consideration and OUI main results. In Sec-

tion 3 we quickly review and elaborate on the theory of cornplex deforma- tions. In Sections 4-5 we present our theory of resonance fke energy for abstract systems with finite number of degrees of fieedom. In Section 6 we prove exponentid bounds on eigenfunctions of interest. In Section 7 we per-

form a spectral andysis of the defomed Hamiltonian and, rnoreover, prepaxe the ground for the analysis to folIow. In Section, 8 we undertake our anal- pis of the decay probabiiity P(t) and prove relation (1.1). In Section 9 we prove a quasiclassicd bound (1.3) on the imaginary part of the resonance free energy.

Now we kt some standard notation we use in the text below: t

(2) := (1 + (XI*) ' , (A) , := (a, Au) and

x(Q) = charactetistic fundion of o set SZ c Rd .

The symbol C stands for a constant (different in different inequalities) which is independent of all parameters involved, and in psrticular d h, ,f? arid t .

2 HAMILTONIANS AND RESULTS

In this section we describe the claos of Schroedinger operators which we con- sider in this papa and present our main results. The Schroedinger operators are given by the difFerentiaJ expressions:

extendeci to functions on L2 (e). Here we use dimensionless nuiables, ti is a smaü (dimensionless) pammeter descending from either the Planck constant or fiom the inverse coupling constant and V(x) , the potential, is a real C2 function such that the operator H (or, more precisely, its dosure) is self- adjoint on its n a t d domain.

Now we formulate a set of conditions on the potentials V(x), various subsets of which are required for difEerent results of this paper: (A) (Local trapping) V(x) is C, has a strict local minimum, Say at x = 0,

V(0) 2 O, and limlZl,, V(x) exists and is less than V(0) . (B) (B& asnimption) There exists Ab > Xo = V(0) such that VE E

[ h Y Ab):

where Si(E) and SJE) are codimension 1 disjoint closed hypedaces such

that Si(E) c intS,(E) (inner and auter classical turning surfaces at energy

E). (C) (Exterior andyücity) There is XI E (Ao, Ab) such that V ( x ) is So(Xk)- exterior analytic in the sense that V is a restriction to Rd of a furtction, also denoted by Vy which is analytic in the tnincated cone ra(Ai), where:

for some a, ai > O and a smooth dosed h y p d a c e S in Rd. (D) (Exterior non-trapping) V(x) is exterior non-trapping at energies X E

QTPT 8

[A., Xi] in the sense that for all X E [A., XI] there BLi8ts a S,(X)-exterior (definition is given below) vector field v on Rd and numbers a, b, d > 0,

a' > a such that:

and

wkere Dv is the derivative (Jambi matrix) of the vector field v.

(E) (Technical condition) For a vector field vA(x) as in (D) let 5!7&4 (x) :=

x + hX(x) and ~ J ( Z ) := V (cpe~(x)). By condition (C), bA(x) has an analytic continuation in 6 from R into a strip ( z E @ 1 I h ( z ) 1 5 e) for

some É > O (see also Section 3). We assume that this continuation satisfies the estimates:

d o r m l y in 6 in the strip above. (F) (Eigenvslue gap) For aU Ar E (A,, AL) there exists a smooth potentid VA# : Rd -+ R with the following properties (below 6 and a satisfy 6 1 and a > h):

a. VAt (x ) = V(x) for x E int (S&')),

b. Vx.(x) c) strictly increasing to m with 1x1 for x E ezt(S,(Xf)),

c. VA, is exterior analytic with respect to So(Xf - 6),

d. Eigendues of HA# := -Ti2 A + VA# ( x ) in (O, A) are sepaxated by at le& h - a-

We give the definition of exterior vector field (see conditions (C), (D)).

D a t i o n 2.1 (Merior vector field). Let S be a smooth closeà hgper- surface in LP'. Then a smooth vector field v(x) is called &&or to a surfoce S if it has Me following propertàes:

0 v ( x ) = O on and mside of S,

Discussion of conditions (D) and (F). For onwhensional systems con- dition (F) - the eigenvalue gap condition - is satisfied genericdy. Indeed, in that case, semiclassically the eigenvalues En = En(h) of the reference Hamil- tonian HAi := -h2A + VAt ( x ) are given by the BohrSommerfeId quantization

condition:

where I (E) is the action at energy E,

with V E a d8SSicd trajectory at energy E. Thüs for the gap AEn := EWI - En we have:

Note that I (E) is the area inside the curve t + (z( t) ,p( t)) where x( t ) and p(t) are the classicd coordinate and momentum. Its derivative, I f (E) , is bounded below and above for our range of energies. Hence we have that:

for some constant C dependent ody on the potentid VAp (x).

FUiallyt we discuss the condition (D) - the exterior non-trapping condi- tion. We give examples of potentials and vector fields for which it is satianed. Assume in the ext (S, (A)) a potential V(x) behaves as:

with c < A. = V(0) snd a > O and assume that a vwtor field v is of the fom u(x) = g(r)% where r = 1x1. Then in this region we have, for any X 2 A,:

1 -a(& - C) I b > O 2

provided a 5 ag or T is sufnciently large in this region. Thus (2.3) holds for these potentids and vector fields with b = ;a(X, - c) > 0.

The operator HAt := - F A + VA'(x) Will be cailed the refemce Humilto- nian. Denote by E~tk, and PAtk, eigenvalues, eigehctions and eigen- projections of HA?, respectively. Without Ioss of generality we cm choose VAt to satisfy:

for some 7 > O.

Refaence Hamiltonians HA# will be used to apprmimate H in various ways (the most important of which wilI occur later in this section) and A' wül be chosen depending on this approJamaticn.

Let S(x, y, T ) be the action of the instanton - the classical partide in the potentid -V(x) (Le. moving in imaginary time) - going from x to y in time 7:

where p(a) is the classicd path in the potential -V(x):

starting at s = O at x and ending at s = r at y. This action srpresseci as a bction of the instanton energy:

Alternstively, due to the Jacobi theorem, A(%, y, E) is the geodesic distance, i.e. the length of minimal geodesic, between the points z and y in the Agmon Riemamian metric:

(the Amon distance). Conversely, S(x , y, T ) is found by mhhizhg the function

with respect to E. In the standard way we define the Agmon distance between two sets, Say

B and C: A(B, C; E). Denote by A(E) the Agmon distance at the energy E between the turning surfaces Si(E) and S.(E):

Similady we defhe (with r = w):

For each B > O we define the energy Eg as s mhimizer of #(E, P). This energy satisfies the criticai point qation:

Con~e~uently, Eg is the energy of instanton of the period fi8 and

is the action of this instanton. Since A(E) is monotonically decreasing and since -2Af(E) is the period,

r ( E ) , of the perïodic orbit at the energy E and therefore -A'@,) = oo, we

conclude thst for P mifnciently large, the function #(E) has a minimum Eg such that Es + A, as p + W. M e r m o r e under our a~sumptions the period r ( E ) deereases as E increases, Le. the function #(E) is convex and

consequently Eg is a unique minimizer. Let TL = r(XL) = be the period of the trajectory at the energy X i .

Due to conditions (C)-(D) we will consider only the energies E 5 XI. Hence the penods T = tip we deal with are bounded as

Le. p 2 a. Next we f o r d a t e precisely the problem we consider. By the Gibbs state

ut inverse tempemture ,û in the well we meaa localized Gibbs date (2.26). NOW we can formulate the problem. The retaining pmbability p( t ) (which

is actually the probability that the particle is still in the well ôt time t ) is:

with x W ~ = x (int (&(A + 26))), characteristic function of int (SJX + 26)), and where p(t) is the density operator at time t, i.e. the solution to the von Neumann quation:

with the initial condition p(0) = p,, where

ZO@) = &,,Sri e-flE~~*. Note, h d y , that the solution to this initial d u e problem is (H is tirneindependent):

Now we are ready to formulate our first main r d t :

Theorem 2.2. Assume conditions (A)-(F) on the Hamiltonian H . To sàm-

plify the statement we also aasume that /3 is suficientty lawe su that technical condition (8.62) of Section 8 is ~atisfied. Let I' = r(P) := -2lrn(F(P)) wherc F(P) b the monance f i e energy intmduced in Dejinition 4.5 (Section 4). Then:

moddo 0(h-4A2) + O ( ( ! ) O 0 ) and with = i' (1 + O(A)). Here A is an eqonentiatly small (in ;) quantity to be specijied in Theorem 8.1.

Our second main result is:

Theorem 2.3. Assume wndàtiow (A)-(E') for the Hamütonian H . Then:

wheze Sg i s the action of the instanton of "penod @3" (see Eqn. (2.22)).

Theorems 2.2 and 2.3 are proven in Sections 8 and 9, respectively. In Section 11 we present the flow disgram oi the proof and the list of

symbols used. Remark: There is a statement analogous to Theorem 2.3 in physics l i t en t u e whüe there is a non-ngorous derivation, using Feynman path integrals, of asymptotics of a quantie closely rdated to r(@).

DEFORMATION FAMILY FOR

Now, we construct a deformation family for the HamiItonian describeci in the previous section. We will use H d e r type (infinitesimal) deformations. Given X E [A,, Xi] let be a &(A)-exterior vector field on p. Let (19 E R) cpe(x) = x + h A ( x ) . Dehe a onepatameter group of unitary operators U' on L2(R') by:

Due to condition (C) He* has an analytic continuation, in O, into a neighbor- hood of O, R, and the resulting family (for which we keep the same notation HeA) is an analytic fsmily of type-A in the sense of Kato (for proof see

[HS2,Corollary 18-51 and Appendix). Then, theory of spectral deformations and Aguüar-Combes theorem t d

us that (pS2,Chapter 181 and Appendix):

and that the discrete spectrum of HeA, Q ~ ( & ~ ) , represents resonances of H (independent of the vector fieid vA(x)).

Note that the explkit expression for Ifex is ([Si], [HS2, Chapter 181 and Appendix):

where we let ( x ) = V(cpe(x)) and:

ri2 + Tdiv (J' tT ( x ) 3' (2) v in det JB ( x ) )

In what follows we wil l need the semigroup e-BH8*, ,B 2 O. To defme

such a semigroup we estabish the sectoriality property of Hel. Recall the definition of that property:

Definition 3.1. A closed opemtor A is cultecf sectorial if ib numeriai mnge N(A) = {(u, Au)lu E D(A), llsll = 1 ) *P wntained in a sector openhg to the right, Le.

Now, it is a standard resdt of the theory of semigroups ([Pa]) that sectorial operators generate differentiable Co-semigroups. Indeed, the Co-

semigroup property is showed using Hille-Ydda theorem in conjunetion with

the standard resolvent estimate for closeci operators:

for z 4 N(A). Next we prove the sectoriality property of H&:

Proposition 3.2. Let H sotish conditions (D) and (F). Then for IIm(0)I suficiently smdl, HOA is sectonal wtWIth the sectoc

where C = 2max(M,C2), M = 4sup, I I D u ~ ( x ) + D V ( X ) ~ ) ~ fl1.11 is the opem- tor nom of a matriz).

Proof. Subindex X is k e d and omitted fiom the notation in this proof. Since Dye = 1 + BDv we have (a = I m(B)):

A i 1 (x) = D & D ~ = (1 + dhT) (1 + laDv)

= 1 + KY (DU + DU=) - Q~DV*DU

and so ArL (z) is invertible for s m d la 1 . Next let A = 1 - ~ D U ~ D U , B = DuT + Du, K = A - I B A - ~ . Then:

L

where G = ( 1 + dK2)-5 A-;. For la( = Ilm(0)I Sufficiently small:

Le. -aM1 5 Im(Ae(x)) aM1 where M = 4mp, llB(x)ll-

Since Re (%) 2 -Cl, 1 Im(h) 1 Cz la1 and go is bounded by ip- ~e (Ag) p (cf. (3.5)) we have:

Q=

and

Hence:

which proves the proposition.

By a standard result (see [HS2,Theorem 18.61 and Appendix), M a r to

the second part of Proposition 4.4 below, the isoiated eigenvdues of HeA are independent of the vector field vx used and therefore are independent of A:

z. are andependent of X . (3.9)

4 RESONANCE PARTITION FUNCTION AND RESONANCE FREE ENERGY

In this section we introduce our key concept - the resonance free energy. To motivate our approach we review first the standaxd case of a quantum

system in a confining (real) potentid V(x) , Le. V ( x ) -+ m as 1x1 + oo and is bounded below. In this case the Schrodinger operator H = -h2A + V(x) is self-adjoint and the heat semigroup e-Brr is of the trace-dass for /3 > O. As it is usud in dealing with systems at positive temperature (again, to juste

what follows one brings the paxticle in contact with a thermal resewoir at the given temperature), we htroduce the (quantuni niechauical) partition function as:

AU thermodynamic quantities csn be expressecl in terms of Z(P). Of particular interest to us is the quantity known as the (Helmholtz) free energy at temperature T = p-' which is d h e d as:

It plays the role of the ground state energy for open systems (i.e. it char- . -B I acterizes the stable ecpilibrium) , while the Gibbs state pe = plays the

role of the grotmd state. This interpretation is supporteci by the following theorem (for a proof see [G JI):

Theorem 4.1 (Feynman-Kac). Let H be a sev-adjoint opemtor, bounded

beiow with purely dismete specbm. Then as B + cm,

i e . as tempemture goes to zero the free eneqy conueryes to the p u n d state

e n w of the sysfen. In fact something stsonger is true:

where Po is the spectrai pmjection of H on the eigensgaa assoeioted with Eo. Thus the Gibbs date pg wnueryes tu the pute ground state Po.

Now we consider an abstract framework for the situation in which a par-

ticle in qgestion is not confinecl and can (and dom) escape to infini@. The latter proces is characterized by the presence of continuum in the spectrum of the corresponding Schrodinger operator. Consequently, the heat semigroup e-BR is not of t racdaas for any /3 and the notions of partition function and fiee energy do not make sense. However there are situations describeci in the Dehition 4.3 below in which closely related concepts can be introduced. To prepare for this definition we need the following:

Definition 4.2 (Spectral deformation f d y ) . Let S be a strip in @ dong R A one-pammeter group of unitary operators

will be calleci a spectral deformation famüy for H , if the following conditions are satisfied:

a TI,= is a dense subset, D, of the dornuin of H, D(H), which is invari- ant under U' for al1 0 E R,

The famtly Heu = u~Hu','~, Vu E E, has an andytk continuation, from 9 ml into the stnp S, and D is a mre for He, 0 E S (hm now on He, 19 E S, wi[l akro denote the closure of the opemtor above c*r

defined frrst on D),

For al1 8 E S n a?: g ( H & ) c Le. the spedrum of H, LP in the dosed lower ho[f-plane.

Next, we consider bounded potentials converging to some limits at infin- ity. Introduce the monance projection:

f

where the dosed, simp1e curve ~ ( 0 , E, d ) is mch that:

7 ( e , E, 8) c p(H8) (the redvent set of He),

7(B, 4 8) = 7+ u 7- u ro with:

where Vm = IimlzI, V ( x ) and Vm = sup V(x) and int(7) designates the interior of the region enclosed by a simple closed m e 7.

In essence, what the resonance projection will do is to project to the (complex) eigenvalues of He whose imaginary parts are at the most E (in absolute value) and the red parts are above the minirnum of the continuous spectrum and less than Vm = sup V(x) . We wiil see in Section 7 that for He described in Section 3 there are no eigendues in a strip hN 5 Irn(z) 5 ch (for some c and a laxge integer N) and therefore resonance projection d not depend on E as long as hN 5 I 5 ch.

Now we are ready for:

D a t i o n 4.3 (Resonance partition function). If for a @en H a d t o - nian H there is a speetml defornation family U = {Ue 1 B E B) and an open set S2 C S n C'+ such that ë p H e ezists, then define:

Z(@) uill be called the resonunce partition function of H . By the definition it is a wmplexfùnctàon of 0 and H and it depends on E as a parumeter. It is

shown below that it is independent of O, pmvided He is o sectorial and type-A analytic opemtor.

To derive some key properties of the resonance partition function, Z(P), we need an explicit representation for the heat semigroup e-@He. A secton- ahty property is sufficient for çuch a represennttion.

Let C be a smooth m e , in the complement of the Hrsector, ninning kom me-SV to me" where <p = 5 - 9 (see figure below). Then for 811 p > 0:

e - B g ~ ( t , ~ e ) d z , (4-6)

in d o m topology ([Pa],Th.1.?.7).

With the p r e I i m i n e behind us we proceed to the main analytic r e d t of this section.

Proposition 4.4. Let U = {& 1 6 E R} be a spectd deformation famity for H in S and let He be the comsponding deformation of H. Assume that Uim is an open set R C S such that He, 6 E R, is sectonal and analytic tgpe A in the sense of Koto. Then the funetion Z(P) := TT ( ~ ~ ~ ~ e - f l ~ ~ ) YI independent of 6. Moreover, the monunce partition function i s independent of the spectd deformotion fumily used, as long as the above conditions on He ow satisfied.

Proof. Since He iS malytic type A in 6 E a, i.e. (He - z)-' is an malytic operator function, and by (4.6) so is ë f l H ~ . This implies analyticity of Z(p) = Tr ( ~ ~ . ~ e - @ ~ @ ) for 0 E Cl . hirthermore, by the definition and whig (4.6), for all t E R:

and so Z(P) is independent of Re(@) (by the cyclicity of the trace). Therefore being audytic it is iridepeudent of 19 E Q.

On the other hand, the independence of resonances on the deformation family is a direct cansequence of ([HSZ],Theorem 16.4(3)). O

Due to equation (4.7), the set C l can always be considered to be a strip.

With the above definition of partition function we now define the reso- nonce jke en- for R as a correspondhg fkee energy:

Definition 4.5 (Resonance fhe energy). We define the resonance energy as:

where we take the principui bmnch of the loganthm.

5 FEYNMAN-KAC THEOREM FOR RESONANCES

In this section we show that the resonance free energy converges to the r e m

nance eigenvalue corresponding to the &round state as the temperature goes to zero. To do this we extend the Feynman-Kac theown to the resonance case.

Theorem 5.1 (Feynman-Kac for resonances). Let He be seetorial with angle d > O . If there is only one (possibly degenemte) eigenualue zo satisfilang:

i. e. monunce fsee enertjy converges to the ground stute resonance eneryy of the system ot zen, tempemture ( t h m is no true ground state sance system U

unstable). Moreouer, (in unifonn topology) :

where p e ~ = Z ( P ) - L P ~ ~ ~ ~ - ~ O and Po. is the pmjection onto the eigenspace of He corresponding to the eigenvalue s.

Proof. Let od(He) = {%ln E Z+) with a ' s ordered so that they are non- decreasing in the real part and repeated according to their multiplicities. By the dehition of P8C8, it and therefore ~ ~ ~ e - f l ~ @ are finite rank operators. By s standard Cauchy argument:

where m is the multiplicity of s, i.e. rn = Tr(Pgo), and

Therefore, since F(P) = -b [-pz, + ln(m + R(P))] , we have (5.2).

Next, we proceed to proving relation (5.3). Let r, be a contour around an isolated eigenvalue z. of He. Define:

Denote Hi = HHe - 2,. Now let r be the c u v e (see figure below):

where c. and c* are the irnaginary parts of the intersection points between C and the line { ~ e ( r ) = F} in the fourth and the first quadrants, respec- tively.

Deforming the contour of integration in (4.6) fiom C to r, U î and using (5.7) for n = O, we find:

By the choice of the contour r, r c C\N (Hé), and by the standard es-

timate llR (2, Hé) 11 5 (dist(z, N ( H ~ ) ) ) - ' , for z $!! N(HL), we know that II R (2, Hé) II 5 C < m for al1 z E r. Therefore:

where in the last equaüty we used that Re(z) 2 F. Thus we have proved that:

which together with (5.4)-(5.6) completes the proof of the theorem. O

6 EXPONENTLAL DECAY OF DEFORMED EIGENFUNCTIONS

In this section we prove exponential bounds on eigenfunctions of elliptic operstors of the form K = p a(x) p + W(x). Such bounds go back to Deift- Hunziker-Simon-Vock ([DHSV]) and Agmon ([Ag]) (see [HS2] for a textbook

exposition). Recall the notation (z) = JG. Theorem 6.1. Aasume Re(o) > O and let z be an eàgenvalue of K with an eigenfunction $. Let S be a smooth closed hypersurface such thct:

on ext(S), for Borne u > O . Then the eigenfunction + (ratkfies:

where 7 2 0, 2" is the churocteristic fundion of the set:

und f ( x ) is the geodesic distance from S to x in the Riemannian metric:

Pmof. First we consider the case of 7 = O. Let xa = 9 ($dist(x, int (3))) where q(t) is a C bc t i on

and such that laql < 32 for s = 1,2. Thm the cut-off fwiction is supported in the domsin { x E ext (S) 1 dist(x, S) > ta).

Let <p = x a e i ~ and ~f := e f ~ e - f . Fiist we want to show:

To that effect use:

Re (Kt)* = ( p Re(a) . p - V f + Re(=) V f + Re(W) + A), (6.6)

where

mations (6.6) and (6.7) with E = 1 imply:

1 t

where, r e d , b = ai + Hence:

on the support of xa, which is satisfied for f and z given in the theorem (see

below) . Now (6.4) follows fiom equations (6.5) and (6.8), providecl (6.9) holds. On the other hand:

where we have used that f = O on supp(Vxa). Since (@p ( 5 32a-pI and since:

Equations (6.4), (6.11) and the definition cp := f t(i impiy (6.2). The only

thing left to prove is the statement (6.9). Fust observe that by condition (6.1), the right hand side of (6.9) is non-

negative on ezt (S) . Now, let f(y) := p(z, y) i idTEP, L,,(r) where L,(y) is the length

of the m e y in the Agmon metric for the potential Re(W) and energy E = Re(z) + v. Then the length (in the Riemsnnian metric defined at the beginning of the theorem) of m e 7 is:

whete #(x) = (Re(W(z)) - E)! and 1-1 stands for the Eudidean nom in Bd.

where (r) = (1 - T ) y + r (y + rb) h) . By the definition of L(7.) we have:

The last two relations give:

1

Since #(v) = (Re( W (y)) - E) ; and since E = Re( z) + v , the last inequahy implies (6.9), which completes the proof of the theorem for 7 = 0.

NOW we consider the case 7 > O. We defme:

~ f ; r := ef (z)~ K (2)- ëf ,

where, recall, (z) = JIflzlZ, and

Then, since V (x ) = (x) )- x:

which for small enough h implies:

1 Re - z)y 1 IIvI12 3

for cp the same as above, provided (6.9) holds. Rom here the proof is parallel to the one of the case 7 = O. Indeed, continuhg from the (6.10) :

where M = s ~ p , , ~ ~ ( ~ ~ , (y) and we, again, used that f = O on supp(Vxa). Ci

We apply Theorem 6.1 to the operators HA#, the reference Hamiltonian introduced in Section 2 , and HAte = U ~ H ~ ~ U ~ ' , the deformation of the refer- ence Hamiltonisn using a S,(X)-exterior vector field VA (as in Section 3).

Theorem 6.2. Let A* < X < XI and 6 > O and let EAtn be an eigenvalue of HAt. Then the corresponding eigenfunction, +Atnt autà&es:

whe* 7 > 0, x is the damderistic funetion of the set ext ( s~ (EA~. + )) , and f,(z) = AM (Si (EAtn), x , EArn) is the (VAt) Agmon distance, at energy

EAtn, from Si(EAt,) t~ Z.

P m f - Denote E = Epn, V' = VAt and A' = AAt. We apply Theorem 6.1 wi th K = HA,. In this case we have a = 1 and W(x) = Y'(%), which implies b = 1 and dsws = dsr, the Amon metnc for V' and E + v.

For an eigenvalue E = E A p n of H't , take S = Si(E + v) and

Then Theorem 6.1 with S = Si ( E + u) implies the estimate

where Fm(%) = At (Si(E + u), x, E + v). We show now that:

Indeed, by the triangle inequality

At (Si(E + v), Si(@, E) 5 Lho) (6.17)

where y. is a straight interval fiom Z' E Si(E + v) to x E Si ( E ) and

1

where #(y) = (Vt(y) - E):. Use:

Y = Vr(xr) - Vr(z ) = VV'(2) (x' - x ) , (6.20)

QTPT 32

for some z in the intenral between x and Y. Take d such that (z' - z) is parallel to VVf(x). Then:

12' - X I 5 CU . (6.21)

So, (6.17), (6.19) and (6.21) Mply:

A' (Si(E + u), Si(E), E) 5 C(V)V . (6.22)

Findy we have:

The last three inequalities imply (6.15).

Now we estimate a. Let x E Si(E + u) ûnd XI E Si(E + 2 4 . Then,

where

= int (Si(E + u)) n ext (Si(E + 2 4 ) . (6.25)

This implies:

Next we estimate IwAtRll. Since V1(x) 2 A. = V(O) by our construction, we have:

HAp +v' 2pZ +A,

This leads to the estimate:

T h now = fbl. Then esthates (6.14), (6.26), (6.15) and (6.27) imply (6.13). Cl

Let now HAte be the complex deformation of the reference Hamiltonian

HA) with a vector field V A exterior to the surface { x E Rd 1 V+&) = A) where X = A' - 6. Here 6 is the same as in condition (F) of Section 2. In case of K = Hxo we have:

Theorem 6.3. For any b > O, the eigenfunctzon, of the opemtor Hpe,

comsponding to an eigenvalue EAt, satl9fes:

where 7 > 0, x Ls the ~hamcteristic function of the set ezt Si(Exn + 61)) ( und f,(x) is the geudesic distance from Si(EAfn) to x in theb Agrnon metric for the potential VA# at energy EAr,.

Proof. The operator HA$e can be expressed as in (3.4)-(3.6) but with the potentid = V 0 cper replaced by V , . Thus it can be identifieci with the operator K if we set a(x) := A&) and W(x) := go(%) + Vu (x). Now we express the m a t e b entering (6.3) in terms of the vector field v = U A

(A = A' - 6). We calculate:

and W ( x ) = b A ( x ) + ge ( x ) . NOW, if 6 = icp:

and letting A = 1 - gDvTDu and B = (Du + DaT):

QTPT

and

Then a = 1 and go = O in int (So(X)) and therefore f (x) is the (V, EArn + -1- Agmon distance fiom S to x for ail x E int (S,(A)).

The rest of the proof follows the lines of the proof of Theorem 6.2. O

Let zwXI, be the characteristic function of ezt (S, (A1')). Dehe:

where 7 is the same as in (2.12) and A" 2 A' > EAln. This hct ion is used to

estimate the difference bA - VA/ of potentials and certain cut-off functions a p plied to PA/, (see Section 7). Let furthermore A,, := A (Si (EAt,), So(EA1,), &,).

Proposition 6.4. Let XIr 2 A' 2 Ep, and B := A'' - Ep,. Then:

Prvof. Denote E = EAtn. Since VAt = V on int (So(X)) and Xi' 2 A', Theo- rem 6.2 with b = IiZ implies that:

A ( s ~ s - ~ ( A ~ ' I S ) &,pu 5 ~ t i - ' e - (6.31)

Next we show that:

Indeed, by the triangle inequality

S i a r l y to (6.22) we obtain

so (6.32) fo11ows. Now, (6.31) and (6.32) and the notation A,, := A (Si(E), SJE), E) b p l y

(6.30). O

hi a similar fashion we prove the estimate for:

where PAld = u ~ P ~ ~ , , u ~ ' .

Proposition 6.5. Let A', A" and tY be os in Proposition 6.4. Then:

7 SPECTRAL ANALYSIS OF THE FAMILY Ha

Let H = -li2A + V(x ) and V(x) satisfies conditions (A)-(F). In this section we conduct a spectral analysis of the operator HOA, h ( 9 ) > O, which is the spectral deformation of H introduced in Section 3. To study eigenvalues of Ho = Ha in the strip

A' > A, we compare He with HAt. Our derivations are aimilar to those of [Si21 and [HS2].

Denote eigenvaiues of He by s. Also, let M be the lowest index such that EM > Ai i.e. M = min{n IE, > Xi).

Recail the notation: HOA is S,( A)-complex deformation of H, HAt is SO(X')- potentid deformation of H (see Condition (F)). Let A' 2 A, 6 = 19, cp > 0. Now we have:

Lemma 7.1 (Bound on the reaolvent). Assume conditions (B)-(D). As- sume numbers a und cp = Im(0) sut&& the inequdity

where b is the same as in (2.3). Then the relations:

imply:

with C independent of a, A, X and ti.

P m f . In the proof below the parameters A and A' are fixed and we use the simplified notation Ho HOA and HI = HAt. Pick a partition of unity

ci*= 1 such that

and

The functions X j can be chosen so that VE > O:

For example ;yo(z) = x (& (V-(x) - A')) where ~ ( s ) = 1 if s 2 1 and = O if s 5 0, and V-(x) is shown on the figure above: it coincides with V(x) in

ext (So(Ab)) while it is monotonically increasing in int (&(Ab)). We asmune that:

Usingthat II(Ho - Z)$I(~ = llu(He - z)+(12, thencommuting~ through Ho - z and using the triangle inequality we obtain:

Ushg that Hg = H = H' on supp(~,) , we find

We will show below that:

for some smooth bounded Function q sa t img:

and

Next, using that

we obtain

where is a characteristic fnnction of sz~pp(V%).

Now using (7.6) and (7.7) we derive:

Then equations (7.9)-(7.17) imply:

and therefore:

Thus we obtained the foIIowing constraint on and bp:

6" w (a2h) and b p W

Estimate (7.19) with (7.23) irnplies (7.3). Thus it remains to prove (7.1 1)

which was as6umed so far. We proceed to proving (7.1 1). Let u = xi$. Note that u is supported in ezt (S, (A")),

~ i c k a parameter 6 satisfying

We dehe a new partition of anity (jk}t, ~ k j i = 1, having the following properties:

and

supp(jo) c int , jo = 1 on int

Next we estimate the pieces, starting with jo:

where Te := = He - b. Now:

(where the second term cornes fiom go). Next we show that:

on int (s~((x + i8)) n ezt (So(X)). Indeed, denote 52 := int (S. (A + ib)) fi ext (S.(Y)). By the definition of n:

and by our assumptions for cp sufEciently md:

Hence by the Taylor expansion to the second order we find on R for <p mifn- ciently smd:

which implies (7.31). Equation (7.31) and the condition Re(z) < X imply:

provided

On the other hand:

where the term cornes fiom Im(ar) and 1 comes fÎom the double commu- tator.

Now (7.29)-(7.36) (with j = O and e = f) imply:

provided

Next, we turn to the k = 1 term on the right hand side of (7.28):

Using the relation:

and condition (2.4) wïth X = Xo + tb, we obtain:

where 7 = a' - a > O and the last term on the right hand side cornes Eom the gpterm in Td, provided p is SUfEiciently s m d .

N a ,

on ext (S.(& + ib)) rn supp(,ji) > supp(u), by non-trapping condition (2.3) with X = A, + ib.

Equations (7.41) and (7.42) imply:

1 1 -In@+ tpa) (He - z ) ~ , . 2 S ~ ( l p j i ~ 1 ~ 2 + (@+Wz)) I I ~ U I ? 9

provided cp is SUfficiently small and

Now we use the estimate

and eqnstions (7.39), (7.43), (?.a), with c = $py, to get:

1 1 h2 IWO - +II lli:~I1> p I I P ~ ~ U I I ~ + ( p b + ~ 2 ) ) 11iiu11~ - C- 1 1 ~ 1 1 ~ . a 7 p

(7.46)

Combining (7.28), (7.37) and (7.46) we obtain:

provided

which folIows fiom (7.24), and

1 h ( z ) 2 --<pb 10 . (7.50)

Using that for sny function jk such that h m L on supp(Vjk):

I I P ~ ~ U I I ~ 2 II~~PII* - c ( ~ l ) ~ 1lkull2 (7.5 1)

with k = O and wing that

for all a and takùig here

1 - I a = 8 min(& vb) = p b (7.53)

Estimate (7.54) impiies the desired estimate (?.LI), if we take

which, due to (7.4) and (7.26), holds if

However we cm avoid this condition if we use &O the iw 1lP j,ul12 term as

in (7.51) with k = 1. Thus (7.11) is proven and with it the lemma. O

Remarks: (1) If we do not use (?.?), i.e. if we use instead of (7.17) the simplifiecl estimate:

then we obtain instead of (7.18):

This yieids (7.17) but under the conditions:

The 1 s t restriction is much worse than (7.23). (2) Estimate (7.45) can be improved if we use the relation:

Recall the following quantity, introduced in Section 6, (which is exponen- t i d y small in h due to estimate in Proposition 6.4):

where XwB is the chsracteristic function of the set ezt (S,(E)). As was mentioned above, the function A,,wu is used to control the p e

tential ciifference WoAt := bA - VA? and certain nit-off functions. Observe that WeAt = WON x (So( A')) , provided A' 2 A. Here recall x(Q) deuotes the characteristic function of a set Q. Hence by condition (2.12):

Lemma 7.2 (Grouping of refmence eigenvalues and tesonances). Let X 5 A' and let a sutii~fy condition (7.1). Assume that dh-2Adtx 8: 1, for ail n sudi that EAt, < Xi. Then:

P m f . In the proof below the parameters X and A' are fixed and the shorthaad Ho G HoA, EI' = HA:, PA = PA#,, WB>( = Wb and En = EAt, is used. Let r, be a contour around En of the diameter k, with a satisfying (7.1) and such that dïst (r,,,o(Ht)\{En}) 2 k.

Now, since we need to prove that Es has at least one eigenvslue in int (î,) we assume the opposïte (that there axe no eigenvalues of Ho inside of it) and

derive a contradiction. Indeed:

Since We = O ( k t X ' ) and 1 z - En 1 = k, and due to tesolvent estimate h m Lemma 7.1 we have:

which, due to the assumption &F2AdtW < 1 contradkts the fact that

En E int(r,). This completes the proof of the Lemma 7.2. 0

By condition (F) of Section 2, there is a contour r, around the eigen- value Ex,, which is at the distance at least & from a(&). Let PAtn be the eigenprojection of HAt corresponding to the eigenvslue EPn. Then according to Riesz:

On the other hand by Lemma 7.1, r, c ~ ( H W ) . Hence we can also define

which, in prinuple, ean be zero, but which is not as we will show later on.

Lemma 7.3 (Reduced resolvent estimate). kt (7.1) hotd for some a = O(1). IRt X 5 Ar < A'' &th X' - Ar > h i . Assume tr2A,,px a 1. Then for dl EAtn < Al:

Pm$ In the proof below the psrameters X and X are k e d and abbreviations He E HaA, Ph = Ph, H' = HX' and PU, = PXn are used. Here Peh and PA#, are eigenprojections for HOA and HP, respectively. Let x,, be the partition

of uni@ described in the proof of Lemma 7.1 and &art with (almost) the same estimate. However, since we are interested in reduced resolvent we will look at the behavior of the resolvent on the &n(Ph) where Fe, = 1 - Ph. As in (7.9) we derive

- where J, = Pen$ and Te := Ho - = p - Ae p +go. Now let us analyze the pieces. Since Lie = Hxr on supp(~o)t we have:

Since HI = HJ on supp(xo), we fhd that H'x0 - xoHo = [Ht, x0] and th- fore:

Since I?,, is at the distance % fiom En, we c m apply Lemma 7.1 to (7.71) and use that V"xo is mipported in ezt (So (A")) (se (7.4)), with X - A' > h f ,

This together with (7.70) yields

This together with (7.69), (7.11), (7.17) and the condition h-*AnXItu Q: 1 gives (see &O (7.8)):

The latter inequality implies (7.68). O

RecaIl that qAn is the Riesz eigenprojection of IfdA dehed in (7.67) and PA/, be the eigenprojection of HA# corresponding to EAtn (see &O (7.66)) . Proposition 7.4 (Closeness of projections). Ld X ( A' < A'' and X be

supported in ext (SJX)) and x = I - z. Assume fr2AnAtAt < 1. Then for

011 Ep, < Xi :

Etrrthermore, PA', can be eueryruhere replaced by the deformed teference pro- jections = u ~ P ~ ~ ~ u + ' toith the estimates m a i n h g the same, pmvided

A' - X 2 6, where b is the same as in condition (F).

Qm 50

PmoJ We use the notation He = Hex. Let us fkst prove the statement (7.75). using

we obtain:

(P0h -

This equation (7.75).

equations (7.66) and (7.67) d the second resoIvent identity

together with estimates (7.3) (Lemma 7.1) and (7.63) implies

To prove (7.74) we use equation (7.78) and use the second resolvent qua-

tion:

to obtain:

that both resolvents are bounded as in Lemma 7.1 we have h d y :

This proves (7.74). Equation (7.76) follows kom (7.77): XP - PX = - XP. And equation

(7.77) folIows fiom the definition (7.62)- The second part of the proposition is proven in exactly the same way. O

h Proposition T A implies in partidar that a dise azound EAtn of radius , contains eigenvalues of HOA of the total multiplicity not less than multiplicity of EArn

FinaDy we mention that as in [Si2,Proposition 10.31 or [HS2,Eqn.(20œ55)] one can show that for the Riesz projection defined in (7.66) and (7.67):

Consequently, the total multiplicity of eigenvalues of Hex inside the disc, D (EA~,, &), around EAtn of radius 2 is equal to the multiplicity of EAfn.

Rom now on we amime that the eigmualues EAtn are simple. Hence inside the disc D (EAt,, $) we have exactly one eigenvalue of Her which we denote by h.

Proposition 7.5. Assume that the conditions of Lernma 7.9 hold. Let n be such that f i t n < Al and let z, be the eagenvalue of HOA in the disc 1 - E n a . Then:

where the corntunt C is independent of h.

Pmof. We will use the abbreviations He = HOA, H' = HA), P;: = f i n ,

Ph = Pan, En = Ep, and A,, = buu. Let ïn be a circle around En of radius 1% - Enl. Then since there is no spectnim of Ho inside rn:

P. = & frm [R (2. Hi) - R (1, Hg)] dz

where, in the last h e , we used residue calculus and Laurent expansion of R (En, He). This implies:

Now by (7.63) and (7.75),

Therefore, using Lemma 7.3 and, again, (7.63):

The 1st inequaüty and the condition h%,, < 1 imply (7.80).

Theorem 7.6 (Estimates on the resonance eigenvalues). The pamm- der A' can be chosen so that X' 2 EAf, und

P m f . TO begin with we take X = A' 2 EAfn. Pick A" such that 1 > XIf -Af > hf. Then by Proposition 6.4, ti-4Ad~1w <s: 1 for li d c i e n t l y small. Cons* quently, the conditions of Proposition 7.5 are satisfied and therefore estimate (7.80) holds. Combining this estimate wit h estirnate (6.30) of Proposition 6.4 with P = A', we obtaiu

where B := A' - EAf,. Finally, we push A' = X exponentiady close to EAtn. We can do this since

by (7.82), EAtn changes exponentially (in l/h) little when A' changes on the scde O(1). So given Ex, with X = Eg, we fint take A' = Eh and then adjust

it exponentidy little to have A' 2 EAtn and A' - EAf, 5 e-* for some e > 0.

This rnakes CS = O (e-f) so that the factor e q in (7.82) uui be sbsorbed into a constant. Hence (7.81) results. O

8 TUNNELING PROBABILITY AND RESONANCE FREE ENERGY

Our goal in this section is to prove Theorem 2.2, Le. to estimate the proba-

bzty that particle initially localized in the well of the volcan&ype potential at temperature T escapes through the barriet. We use the dehitions and notation from Sections 2-4. Let XI - 6 2 X > Es and A' = AL, where XI and 6 are the same as in conditions (C) and (F) of Section 2, respectively. Note

that A' 2 X + 6. Let I, and En be the eigendues of the operators HOA and HAf, respectively, and în = -2Im(%). R e d that the eigendues, a, of H& are independent of A.

More proceeding with the results of this section let us define the following - quantities: ï = Ch,,, pnrn and ï@) = c%<~, A (rn -q2 ~ i t h =

(C&<Ai e - ~ & ) - L e-86,. The fmt main result of this section is the following:

Theorem 8.1. Assume conditions (A)-(F) on the phys id Hamiltonion H. Then with the definitions &en in the preceding pamgmph rue have:

h o0 modulo ~(h-'A2) + O ( ( T ) ) . Here A2 = pnAkLAl.

Prwf. In the proof beiow the parameter A' is Exed as in the statement of the theorem. Henceforth we use the foIIowing abbreviations H' = HA#, Pk = Px, and An = First we prove the following intemediary proposition:

Proposition 8.2. We have for dl t E w:

Pm06 Firat note that:

Here A and At are closed intervals satismg En E At CC A and Ano(Ht) =

{En}*

where

= TT ( x e - 8 c ~ v e * ) . (8.8)

QTPT 55

Sinee P'g(Hf) = O , we have P"g = P f (g (H) - g(Hf)) . Using the Helffkr- Sjoestrand operator calculus (see [DiSj], [HuSil) we write:

where ij is an almost analytic extension of g and 6 = a@& dy) and similady for g(Hf) . Thur and the second resolvent identity give:

Using this formula and (7.63), we h d that :

Hence we obtain

Now we estimate the term:

Estimate (8.11) gives ri& away that Agg = O@). To obtain a better esti- mate Agp = 0(A2) we have to work harder. First we t r d o r m by the Stone theorem:

There Btists an ah& andytic extension j ( z ) of g(A) into @ satisfying:

(see [DiSj] and [HuSi]). By Green's theorem and since (H - z)-~ has no poles in C (cf. [BZj) :

where we have used that:

for Im(z) < O. This talces care of the second term on the right hand side of

(8.13). Now we analyze the first term on the right hsnd side of (8.13). We use

the complex deformation for the term under the trace. Let UA be an S,(A)- srterior vector-field satisfying conditions (D) and (F), which was used for the definition of HOA, and let LTe = Uex be the corresponding deformation group. By conditions (C) and (F) the operators HeA and Pi, := LT'PU;' have analytic continuations in 0 and in 0 and a, respectively, as typeA f d e s . Denote by Ph = PoA, the eigenprojections corresponding to G. As above, when it does not cause a conhision we omit the subindex n: Ph = Po, and &O e t He = HeA.

Now for 0 real we can insert the operators u;'u@ = 1 inside the trace

and continue the result in 8 to obtain:

for Im(0) > O and X E supp(g) , where we used that X ~ g L = x since V A = O

on supp()o, and where we removed f10 from the resolvent Qnee R (A, Ho) is analytÏc in X E supp(g) .

We estimate the integral:

R (A, &)e-Yg(X) dA .

Define a domain R enclosed by the r d axis and a smooth m e I' = P ++Il

where (I(A) is an open interval containing A):

r" = ((x, -E) 1 x E A) (8.19)

and r1 is as in the picture below:

Ushg Green's theorem for the domain 0 and using that

a. f i , Im(9) > 0, has no spectrum in n mpp (a@) and

C. II R (2, He) 11 I: C a R L on I"' and on supp(@) by Lemma 7.1 since the &stance from r" ü supp(@) to a(HAt) is greater than

we find that

where Pej is the eigenprojection of Ho corresponding to y E O#&), Im(6) > O. The second terni on the right is O (ë*) and the third term is O ((4)-) as above. We csrn assume that:

Using (8.13) and (8.18) and that X& = x we obtain:

Thus by (8.16) and (8.20), (8.21) we have:

Now we estimate B, using Proposition 7.4 and (8.11) (see &O (8.35)

below) :

where C := Tr ( X ~ $ , , g e q ) . Now using that &, = XP and inserting a

cut-off function g, = gL(H) such that g,g = g and g(A) = O near X = En, we obtain:

We write C as (in estimates below we use Proposition 7.4 and (8.11) with g and with g,)

Hence C = O (A2) and therefore B = O (E2A2). This implies by (8.23):

Equations (8.7), (8.12), (8.28) and (8.29) imply:

NOW we investigate the t e m A,. We write using the Stone theorem as before (see (8.8)):

dX d p e-%*i?g(~)g(p)

Tr ( X (R(X + 20, H) - R(X - $0, H)) P' ( R ( p + 2Q1 H) - R(p - 20, H)))

The terms involving R(X - 30, H) and R(p + 20, H) contribute O ((4) -) as above. Also as above we deform the terms R(X + 20, H) aad R(p - 80, H) to

where, recaII, Po, = UoPU,-l, Im(0) > O and Im(a) < O. Using the Green's formula as above we find

Here we used that a,(&) = o , (He) . Now we prove the following,

P m f . In this proof we set Pi := Pb. Let X = 1 - X. We write:

Now we apply the foUowing three estirnates of the second part of Proposi- tion 7.4 and definition of A = h, in (6.29)):

Po' (Pd - P i ) P i = O (E31i2) (8.34

Pé(Pe -Pd) = O (E'A) = (Pe - Pi) Po' (8.35)

xP = O(A) , (8.36)

to the tems above in a straightfomard manner to obtain (8.33). Cl

Now equation (8.32), the above lemma and the notation T;, = - 2 1 m ( 4 give:

A,, = ( 2 ~ a ) ~ e - Y (TT (P) + O (E4h2)) + O ((3") which together with (8.30) implies (we redore now the subindex n):

which togetber with (8.3), the convention that the eigendues En are connted together with their mdtiplicities and the relation Ch,, p,, = 1, imply the statement of the proposition. Cl

Now we recd the following definitions:

Note that r is the (truncated) Gibbs average of rncs. Next we use the following easy inequality:

Lemma 8.4. Let z p , = l,p, 2 0 and M = sup, ft'(r,) < m. Then:

PmoJ Performing the Taylor expansion of f (rn) around F:

and then averaghg by pn's we get the desired inequality.

Estimate (8.2) implies that rnodulo O (tia4A2) + O ((!)*):

r t where po(t) = pnë* with pn = ~ ~ ( / 3 ) - ' e - ~ ~ and

Using Lemma 8.4 for the upper bomd and Jensen's ineqnsüty for the lower bound we obtain:

where ro = mi% ri- Ecpations (8.42) and (8.44) imply (8.1). a

Using that the maximum of the second term on the right hand side is reached at t. = we obtain:

for O 5 t 5 T, where r. = niuz $*). Now we are ready to connect the Gibbs average of tunnelhg probabilities, - r, to the free resonance energy F(p) , namdy to r = -2Im(F(/3)) (F(/3) is

given in Definition 4.5).

Themem 8.5. Assume conditions (A)-(F) ore sotisfied. Then:

Pmo/. First we consider the case pl\, 5 e for some E < 1. Let 21 = Re(Z(/3)) and Z2 = Im(Z(P)). We will use the formula:

Now we estimate 2. Since Z(P) = CkCA, e-B*. we have:

and

where we used the definition r, = -21m(h). By Propodion 6.4 the con- ditions of Proposition 7.5 are sstisfied for h SUfficiently small. The latter proposition with A' = Al implies:

where, recd, En = EA,, and 1\, = &AIA1. Using this relation, we h d fui.thennore that

Since r, = O (x) by Proposition 7.5, we have finally that:

Similarly we compute for Z2:

Dividing the latter expression by the former we find:

where recall 5 a < 1 by an asmimption. Now we consider the case of e not too md. Denote Xlo = Re(z1) - Re(&).

We have in this case:

and therefore we have for F(P) = -; ln Z(P),

Since 2 É, this gives

Observe now that Ato = di + o(h) for some w independent of h. Next ~sing the dennition F = pnrn9 where pn = e-BR (CG<Al e-@&)-l,

we derive

where r,, = r n a x ~ , ~ , r,,. Observe that Eo = wrA + o(h) for some w' > O independent of ti (where we used

impiy that: that V(0) 1 0). The last two relations

(8.61)

d Now, taking e = A& and observing that in this case the error term in

(8.61) is much smaller than in (8.56), we conchde that (8.46) holds.

Corollary 8.6. Assume is so large thnt:

1500 moddo O ((g ) 2, + O ( ( i . ) ) .

Proof of Theorem 2.2. The statement of Theorem 2.2 follows from The+ rems 8.1 and 8.5, and Corollary 8.6.

9 SEMICLASSICAL APPROXIMATION

By now we have developed aLl the necessary machinery for estimahg the upper bound on the tunneling probability r(P) = -2lm(F(/3)).

Theorem 9.1 (= Theorem 2.3). Assume conditions (A)-(F) for the fimil- tonian H. men:

where SB is the action of the instanton of peràud Tip (se Epn. (2.22)).

Pm The st atement follows fiom equation (8.46) and Proposition 9.2 below. 13

Ekfore proceeding to Proposition 9.2 we define a constant Ci by the in- equaüty:

where En := EA,, . Proposition 9.2.

Pmof. By the definition:

Using (7.81), we obtain:

where q5(E,p) is defined in (2.20). Since, by the definition, Eg minimhes

#(Et B) a d by (9.2):

The last two estimates together with the inequality:

and the relation SB = t$(EB, 8) (see (2.22)) imply (9.3). O

10 APPENDIX: REVIEW OF RESONANCE THEORY

Spectral Theory: We start with a standard dcrfinition:

DefMtion 10.1. Let A be a closed opemtor on a Hilbert mace 3C. The dis- m t e ~pectrum of A, ad (A), consists of ail isolated eigenvalues of A with finite algebmic rnu&iplicit y. The essential spectrum of A is oeas (A) := a (A)\od (A).

Now we defhe two other notions of spectra:

Definition 10.2. If for X E @ and A a closed operutor on a Hilbert space Yi7 there exkts a sequence {un) (colled a Weyl sequence for X and A) such that l l ~ n l l = 1, u,, +w O and ( A - X)u, +, O then we auy that h belongs to the Weyl spectrum, W (A), of A.

D a t i o n 10.3. If for X E C! and A a closed opemtor on a HiIbert space 31, thm ezkts a sequence (-1 (called a Zhklin sequence for X and A) such that

ll~nll = 1, a m ( % ) C (2 l 2 E Rd\&) und II(A - 4%11 -t 0 n + 00 (where & ia a bal2 of mdiw k ) then we say that X belongs to the Zhislin spectrum, Z(A) , of A.

These concepts can be used for determining essential spectra of certain ciosed operators:

Theorem 10.4. Let A be a closed opemtor on 31 wiU, p(A) # 0. Then W(A) c o,,(A) and the boundary of aess(A) is contained in W(A). Fur- t h m o r e , i f each connected ccimponent of the comptement of W(A) in C contains a point of p(A), then W(A) = oe,(A). The converse olso holds.

Another useful result dong the same lines is:

Theorem 10.5. Let A be a locally compact, closed opemtor on L ~ ( R ~ ) such W p(A) # 0 and Cr(*) is a mre (plw a technid condition, see [HSZ, T h e o m 10.12]). Then i f each connected component of the complemenf of

Z(A) contains a point of p(A) tue houe Z(A) = W(A) = o,,(A).

For relatively compact perturbations of closed operators (in particular applied to Schroedinger operators) we have:

Theorem 10.6. Let T Be a closed opemtor on 7t and A be a relatiuety T- compact operutor. Then o,,(T) = a,,(T + A).

Reéionances: By Ruelle's theorem for Schroedinger operators, to every eigenvalue corresponds a solution of the Schroedinger equation whose evolu-

tion takes place in a ball of finite radius (a bound state) and to the essentiai spectnun there correspond solutions that leave any ball of finite radius in f i t e time (scattering states). Resonances are a special class of scattering states, namely those that stay bounded (in some ball of finite radius) for a long period of t h e . The notion of the resonance is an inherently quantitative one.

We begin with a definition:

D a t i o n 10.7 (Resonances). The quantum resonances of a Scmetiinger operotor H associated uith a dense set of vectors A in the Hilbert space 3L are the poles of the rnemmorphic continuations of al2 matriz elements

($1 R(z ,H)v} , $, 9 E A from (2 E @ 1 I d z ) > O) to {z E C I Im(z) 1 O)*

Now we describe the essentisls of the Aguilsr-Balslev-Combes-Simon the- ory. To that end we make the following assnmptions:

a. H = -A+ V is a self-adjoint Schroedinger operator with domain D(H) and o,,(H) = [O, ao) with discrete spectnun Q ( H ) C (-00, O].

b. There exists a family U of linear operators U8, B E D = {z E C 1 lzl < 11, such that for O E D n R, Uo is unitary and UoD(H) = D(H) for ail 0 E D and [I. = 1. Mhermore, there d t s a dense set of vectors A C 3C such that:

the map ($,O) E A x D + Ilo+ is analytic on D with values in

x;

for 0 E D, UeA is dense in A.

c. We define, for 6 E D f î R, a family of unitary equivalent operators He r u , H u ~ ~ . We asmime that the map B E D + Ho, is analytic of type-A (see "2, Definition 15.81).

d. There ercists an open, connected set 52 C { z E C 1 Re(z) > 0)such that W ~ w n e # @ , a ~ d R - = w n C - #@,andfora l lB~ D+iDf l@C, aeSs(HB) fî R+ = 0. For each E > O, there d s t s a subset n; c such thst for some 0 E D; r {z E D 1 h ( z ) > a) , we have cm(&) na; =

0.

Danition 10.8. The f d l y U satisfying (b) and (c) is ualled a spectral defornation familg for H. The dense set of vectors A is called the andytic

uectors for [le.

Then, we have:

Theorem 10.9. Let H be a self-adjoint Schmeàànger opemtor with apectd defornotion fumily U and analytic uectors A such that (a)-(d) are satisfied.

Thm:

defined for Im(t) > 0, has a merwmorphic continuation acmss into

Cl:, for anv E > O.

The p o h of the continuation of Fw(z) into Cl; are eigenvalues of al1 the opemton He, 0 E LI:, such that o,,, (Ho) n Q; = 0.

These potes ore independent of U.

Proof By asawnption (a), F$&) is analytic on @\R. F i . a E a?. Since, for 6 E R n D, utL = ~ d ; we have:

Now, by (b), Ue$ and Uecp are analytic in O. Furthemore:

Condition (c) assures that R (t, He) is analytic in 19 for all z 6 a(H6). Since for 8 E D we need to adjust equslity U,' = Uè to &' = Ui we get fiom (10.2):

which is anirlytic for 0 E D ônd z 6 o(He). Next, choose r > O and fix z E nt. The function F$,(z, O) , defined for 0 E Rn Dl can now be extended in O into D: by (c) and (d). We fk û E D( according to (d) so that

oess(He) nn, = 0. It follows that F$&, 8) can be meromorphically continued in z Eom $2: into = Q, n C- . Now, since F#,(z, 8) = F$&) for z E Cl:,

the identity principle for meromorphic fimctions implies that there d t s a h c t i o n meromorphic on Q, which equak F*&) on $2:. This hinction is the sought meromorphic continuation into Q;.

Now vue prove that the pole of the continuation of F$&) coincide with eigenvalues of He. Indeed, the meromorphic continuation of Fw(z) into f2;

is given by the rnatrix elements of R (2, Ho) in the states n UeJI and ~8 rn &(o. Condition (b) assures that these vectors, UeA (6 E D), are dense. Thus, if Ho has an eigendue at r\o E 4, F$&) will have a pole there. Convdy, if the continuation d F#&) has a pole at & then it must be an eigendue of Ho.

Independence of poles on 19 and U is now a direct consequence of the

nniquengs of meromorphic continuation. O

Spectral Deformations: As was shown above, the Aguilar-Bslslev-Combes Simon theory of resonmces identifies resonmces of a Hamïltonian H with (cornplex) eigendues of a closed operator Ho which is obtained fiom H by spectral defomations. Here we review a general method of generating these

deformations &g flows on the configuration space [Sil,Hu,Cy] (see [HS2] and [Si31 for reviews).

Let g : R? + RR be a C map. Then, for 9 E R, define a family of maps on IF':

Clearly, for 0 d u e n t l y s m d &je's are invertible. Indeed, denote by Dq50 and Jb = Je the derivative and the Jacobian of 4e, respectively. Let:

Note that 4e is an infiniteaimal version of the global fiow generated by

the vector field g(x). Ezomples: Dilations are generated by chooQng g(x) = x. Exterior dilations

(to the baLl BR(0)) are genersted by:

Next we construct a f d y of unitary operators generated by 4@. For any 11, E S(F) defme:

QTPT 73

Proposition 10.11. The map Uo maps S(R?) into S(R?). For (81 < Mi and d, Ue extends to a unitary opemtor on L2(IPL) and Ue + 1 sfrongly os

6+0 .

Now, we extend operators Ue fiom 6 E R to 0 E C, at least for 101 smd. 1

Rom (10.7) we see thst this extension will be possible if Ji and $ have extensions into some complex neighborhood of P. It is easily seen that the condition 101 < Ml suflices for the extension of the determinant term. On the other hand, for analyticity of @, we need to find a dense set of funcfions in L2(R?) thst are restrictions to IP< of functions analytic on a complex neighborhood of B", and such that tl, o q& remains in L2(R) for 191 < Mt.

Deflnition 10.12 (Analytic vedors). Let A be the lineur space of al1 en- tire functions f ( z ) huving the properfy that in uny conical region Cc,

for uny c > O , we have for any k E N,

iim l z l k I f ( z ) l=O. It(+oo , zECe

27ien the set of analytic vectors in L 2 ( P ) às the set of @ E L'(R) such that 3f E A and$(x) = f ( x ) , XEP.

Lemma 10.13. The set of functioru in A restricted to R" fonn a dense, Ihar &set of L 2 ( P ) . Etrrthermore, for any f E A, f (2) E L 2 ( F ) for

z E Cc and ony e > 0.

More stating the andyticity result we impose another condition on the vector field g(x):

This allows us to normalize g so that Mi = 1. Define:

Then we have:

Proposition 10.14. Let U be a specttul defortnation family associated with a mooth uectorjield g satb&ing (10.8) and ML = 1. Then:

a. the map (8, f) E Do x A + Ue f is an analytic L2-valued function;

b. for any 6 E Do, &A b dense in L2(IPL).

Clearly, as we extend Cle to cornplex 0 unitaxity is gone and thdore Ho = U~HU'' are not isospectral. Thus we next analyze the spectra of Ho. We limit o d v e s to Schroedinger operators H = -A + V(x) .

F k t we notice that:

where pe = t l ep~ë1 and &(x) = V (4e(~)). A straightforward calcdation gives an explkit expression for pi. Indeed, consider first:

Next calculate (using chah d e ) :

Therefore:

and (conjugating for û E R): t 1

( U ~ V U ~ ' ) ' = -Jo5V b ~ & 1 ~ 8 s . (10.13)

Multiplying (10.12) and (10.13) we obtain: 1 1

& = u ~ ~ ~ u ~ ' = J ~ V D # ~ I J ~ D # ~ ~ ~ - V J ~ ~ (10.14)

which c m be put in a useful fom:

QWT

where

Proposition 10.15. The family of opemtors $, 0 E Do, U a type-A analytic

family with domain p(lP).

Pmof. The strong analyticity part of the definition of typeA anslyticity on D, is a &ect consequace of the analyticity of Ag and ge. The invariance of the domain is a bit more involved.

In the rest of the proof we will use the Einstein's summation convention. Let u é p(IP). Then:

and so, by the boundedness of AB and go:

Thus D($) = P(F) c DH). Proving opposite inclusion is more compli- cated. It rests on the bound fiom bdow for 4 (cf. Section 3):

for aii 6 E @< and some c > O. Then, using (10.15), (10.18) and the definition of pe we obtain:

By commuting pk to the left in each term and estimating dl terms we get:

which implies D(& c p(P) and completes the proof. O

Next, the spednun of the deformed Laplacisn 6 is piven by:

Proposition 10.16. Let uector field g satisfar condition (10.8). Then:

for uny 0 E Do.

Pmof. The proof is done by showing that Zhislin, ZM), and Weyl, W M ) , spectra coincide and then cdda t ing the Weyl spectnun (by noting that

go(x) + O and A&) -t (1 + as llxll + m). Since it turns out that dl\Z(& has one connected component, and since that component contains a point of the resolvent set we have that [HSZ,Chapter 101 W M ) = a,,#). Thus, since the disaete spectrurn of $ is empty the proof is complete. O

Now we consider the potential V. In order for analytic extension to exist and to be able to say something about the spectnun of the deformed Hamiltonian Ho, we make the following assumptions on V:

b. V is the restriction to Et" of the function that is analytic on the trun- csted cone Cf, for any c > O and some R > O SUfficiently large.

Then we have:

Lemma 10.17. Let V sot* conditions (a)-@). Then Ve m V + eztmds to 0 E D, as an analytic, rehtiuely $-compact operator.

Pmof- Analyticity follows from andyticity of V on Rcn(40) c Cz (condition (b)) and andyticity of ([Si21,Proposition 5.3). Relative compactness fol- lows fiom the estimate O < q? 5 $ 5 Cg and property of norm-closedness of compact operators. Cl

Corolky 10.18. Let V satish conditions (ta)-@). Then the self-adjoint opemtor He = j$ + L$, d e f i e . for B E Do fl R, &ends to an analytic type-A furnitg of operators on Do uith domain @(R?).

Finaily, putting together the previous results we obtain:

Theorem 10.19. Let U be a spectral defonnation fanaily for the Schmedinger opemtor H = -A + V , with V satLgfYng conditions (a)-(b). Then for an9 e E Do:

(Ho) = { z E @ 1 arg(z) = -2 a g ( l + O ) ) .

Let R(H) be the set of resonances of H and let S i be the open region in the loww half-plane bounded by Plf and a,,,(Hs), for 8 E Do n C . Then

In purticulur, the resonances in S i depend only on H and A.

Pmof. The statement about the essentid spectrum of He = pi + Vg f o h fkom the Weyl's theorem for claed operators since is &compact and:

On the 0 t h hand, the statement for the discrete spectm foilows fiom the Aguilar-Balslev-Combes theorem. O

11 SUPPLEMENT: FLOW DIAGRAM AND NOTATION

The flow diagram of the proof.

Main results: Thgorem 2.2 Theorem 2.3 (9.1)

I Prop 7 . 5 Prop 6 . 4

Prop 7 . 4 Lem 7.3 T b 6.2

Notation.

H is the original (unstable) Hamiltonian.

HOA is an S,(A)-exterior deformation of H, for X [A., Ai].

HA# is the original Hamiltonian H with V deformed at &(A') which is exterior analytic at S. (At + 6).

HAto is m S, (At + 6)-exterior deformation of KAt.

z, is the n-th eigendue of

EAtn is the n-th eigendue of &.

PeAn is the Riesz projection corresponding to the eigenvalue z, of HOA.

PA#, is the Riesz projection corresponding to the eigenvalue En of HA#.

qe8 is the defornation of PA#, Le. = C T ~ P ~ ~ ~ U ~ ' .

0 6 and a small parameters satisfying 6 w ($fi)$ > ti (Section 2, con- dition (F)).

Al the cut-off energy specified in Section 2 (condition (C)) .

= 11 (x)' xwPAt,ll is an exponentidy s m d passrneter defined (and estimated) in Section 6.

Refèrences:

W 1. Atüeck, gUantum-cBlme~iliiçy, Pb. Reu. Lett. 48(6), 1981, p.388-391.

[Ag] S.Agmon, Lectures on exponenticrl decay of solutions of second-& eiiiptic -a-

fions: bounds on eigenfinctim of N-body Schrdinger opemtms, Princeton Math. Notes

29, 1982.

PCl EBslslev, JM.Combes, Spedrol propntics of m a n y M y Schrdinger opemton &th

dilutotion-adyfic interactions, C m . Muth. Phys. 22,19ïi, p.280.

PG] GBiatter, D.A.Gorokhov, Phys. Rev. B 57,1998, p.3586.

pohm] David Bohm. Quantum Zâeory. Dover 1989

PZ] Burg, M.Zworski, Rwonance expamiow in aemic&saM propagation, Preprint, Berke-

ley 2001.

[CDKS] JM.Combes, PSuclos, MXlein, RSeiler, TiK ~ h p e nsonance, Comm. Math.

Phus* 110, 1987, p.215.

[CFKS] H.L.Cycon, RG.Roese, W.Kirsch, B.Smon, S d i n g e r Qmuto~a. Springer 1987

[CL] A.O. Caldeira and A.J. Leggett, Ann, Phy* 149,1983, p.374.

[Col S. Coleman, Aapceb of Spmetry, Cambridge University Presa, 1985.

[Col] S. Coleman, Phys. Rev. D 15, 1977, p.2929.

[CS] Costin, ASoner, Rcsmtonance th- for SchdiBger opmtma, Preprint, Rutgen, 2001.

[Cy] HLCycon, RwoMntm &fincd rnudijkd dücifions, Hdv. Phys. A& 58, 1985,

p.969.

PHSVJ P.De&, W.Hunzi.ker, B-Sion, E.Vock, Pointuiise hndc on eigenfindimrr and

waue pcrckefs in N-body quantum q~stems N, Comm. Math. P h w 64,1978, p.1.

PiSj] Dimassi, JSjoestrarid, S p c h d Asyrnptoties in the SmiCcias&al Li& LMS k-

ture Nofes, Cambridge University Press, 1999,

[EMS] G S ~ o y , L.G. Mourokh, A.Yu.Smimov, Phus. ktt. A 175,1993, p.89.

@SI G.E'Bfi.em09, A.Yu.Smirnw, Sm. Phys. JETP 53(3), 1981, p.=

[Ga] G.Gamoa, Zur gWnkntl>eorie dw A b m k m w , 2. Pw 51,1928, p.204.

IGJl J. GLimm and A. JafFe, Quantum Physics, Springer-Vixiag, 1987.

[ G m H. Grabert, U. Weiss and P. Eanggi, Pht(s. Rcv. W. 52(25), 1984, p.2193.

[HeSj] BJWF'er, JSjoestrand, Ruonance en Limite Snni-clas3i~zce, M m & la S.M.F.

w5, Suppl. & h S.M.F. Bulletin 114,1986.

PSI] P. Ehlop, I N . Sigal. Slicipe Reaonances in ()uontwn Mechanka, in infferdd

Equahow in Mathemclticd Phy~àa, ed. I,Knowles, Lecture Notes in Math. d.1285, Springer 1986

p S 2 l P. Hislop, I.M. Sigal. IntmductMi b Spectrd Tlremy. Springer 1996

pu] WBunziker, LW&ion And#@ and M o l d a r Resmnœ Cumes, Ans INt. H.

Potncum 45,1986, p.339.

PuSi] WHunaker, IM.Sigal, The-dependent scatterir,g UKmy of N-My syatmu, Rew.

MOU Phya., 12,2600, p.1033.

N T. Kato. Perturbation TI,- for Lineor Opemttws* Springer 1966

YuKagan, A.J.Leggett (eds.) Quantum f i n n e h g in Conderued Media, North-

Houand, 1992.

[Knj HXIeinert, Puth Integrda in Quantum Mechanica, Statisttcj a d Polymcr Ph@M,

World Scient&, 1990.

Wall Kubo, Toda, Bashitaume. StcfinicBl Physics 2. Springer 1983

pl] A. Linde, Particle Physics and Inpcitionary Cosmlogy, Harwood, 1990.

Pt] A. Linde, Stochadic appro~di to tunnelhg and unnierse fmation, Nucl. Phva.

B372,1992, p.421.

Fa] J.S.Langer, Ann Phgg. 41, 1967, p.108.

PO] Al. Larkin and k N . Ovchinnücov, Zh Eksp. Teor. Fk 86,1984, p.TL9.

p02] A L Latkin and YaN. Odimikov, J. Low Tmp. Phya. 83(3-4), 1986, p.317.

p] M-Merkli, LM.Sigal, A Time-dependent Uieory of qwntum monunees, h m .

M& Phvs*, 201,1999, p.549.

[OS] YuN.Ovchinnikov, 1.M-Sigal P h p Reu. B 48(2), 1993, p.1086.

pal A. Paay, Sem@mups of Lhmr Openators, Springer-Verkg,

p] M. Reed and B. Simon, Metnwb of Modern Motlimrcltied P h @ s IT, W, Academic

Press, 1975.

[Sil] LM. Sigd, Cmpb t r r r r ~ f ~ i o n method in one-bodg quunfum systerns, Ana Iwt .

61, Poàncure 41,1984, p.10%114.

[SB] LM. Sigal, Shatp Ezponentiaf Brmnàa on Reaonance States and Width of Resonances,

Adu. App, Math. 9,1988, p.127-166.

[Sis] IMSigal, Geomefrical Theory of R ~ s o ~ n e u in Mdtiporticle Syatema, P d i n g s

of iXth Intmofimaf Congms on Mathematid Physies - Stoanaea, Woks, eds. Simon-

T~Iuu-Dw&s, 1988, p.320.

[Si] B S i o n , l'lie Th- of Rwonunew fm Düation A d * P o M & and Me F m -

&ttbns of T h e Dependent Perturhtion Th-, A m . Math, 07,1973, p.246.

[Sa] B.S0mon, I ne Definifion of M o I d a r Resmnce Cumes by the Method of Eztdof

Complez Scding, Phya. Letts. nA, 1979, p.211.

[SJ] BSiimon, Semiclasaical A n d m o f h Lying Eigenuohrw, LI. lùnneling, Ann. Math.

120,1984, p-89.

[Sm] A.Yu.Smirnov, S m Phgs. Semicond. 26(9), 1992, p.933.

[Sv A.Mer, M-Weinstein, hunances, mdidimi àamping and nutobüüy in Hamilto-

nian nonlineur waue eqtrations, huent. Math., 136, 1999, p.9.

m] H.Umemwa, HMatsumoto, M.Tachiki, Thamo Fieid Dynamiw and CondcMed

States, North-HoUand, 1982.

(vS] A. V i i d and E.P.S. Shellard, Comiè St~nga and ofhm Topologirol Defedr, C m -

bridge, 1994.

[zJ1 J.5Justin, Quantum Field Tihcmy and Critical Plimomenu, Mord, 1996 (3rd).