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arXiv:quant-ph/0612027v1 4 Dec 2006

Approximate resonance states in the semigroup decomposition

of resonance evolution

Y. Straussa

Department of Mathematics, Ben-Gurion University of the Negev, Be’er Sheva 84105, Israel

L.P. Horwitzb

School of Physics, Raymond and Beverly Sackler Faculty of Exact Sciences,

Tel-Aviv University, Ramat Aviv 69978, Israel

and Physics Department, Bar-Ilan University, Ramat Gan, Israel

and College of Judea and Samaria, Ariel, Israel

A. Volovickc

School of Physics, Raymond and Beverly Sackler Faculty of Exact Sciences,

Tel-Aviv University, Ramat Aviv 69978, Israel

Abstract

The semigroup decomposition formalism makes use of the functional model for C·0

class contractive semigroups for the description of the time evolution of resonances. For a

given scattering problem the formalism allows for the association of a definite Hilbert space

state with a scattering resonance. This state defines a decomposition of matrix elements

of the evolution into a term evolving according to a semigroup law and a background

term. We discuss the case of multiple resonances and give a bound on the size of the

background term. As an example we treat a simple problem of scattering from a square

barrier potential on the half-line.

1Introduction

Originally formulated for the analysis of scattering problems involving solution of hyperbolic

wave equations in the exterior domain of compactly supported obstacles, the Lax-Phillips

scattering theory1was developed as a tool most suitable for dealing with resonances in the

scattering of electromagnetic or acoustic waves. Subsequent to its introduction by Lax and

aElectronic mail: ystrauss@cs.bgu.ac.il

bElectronic mail: larry@post.tau.ac.il

cElectronic mail: volovyka@post.tau.ac.il

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Phillips, various authors have contributed to further development of the theory2,3,4,5,6. No-

table recent additions were made by Sj¨ ostrand and Sworski7who extended the scope of the

theory to include general classes of semibounded, compactly supported perturbations of the

Laplacian in the wave equation, and by Kuzhell, via the development of a formalism pro-

viding conditions for the application of the Lax-Phillips structure to an abstract form of the

wave equation8and to certain classes of Schr¨ odinger operators9. In addition, Kuzhell and

Moskalyova10applied the Lax-Phillips theory in the analysis of scattering systems involving

singular perturbations of the Laplacian.

Several recent papers have dealt with the adaptation of the Lax-Phillips theory to quantum

mechanical scattering problems. An early work in this direction is Ref. 11,12,13. A general

formalism was developed in Ref. 14 and subsequently applied to several physical models in

Ref. 15,16,17. Such efforts to adapt the Lax-Phillips formalism to the framework of quantum

mechanics are motivated by certain appealing features of the Lax-Phillips theory. One of

these features is the fact that the time evolution of resonances in this theory is given in terms

of a continuous, one parameter, strongly contractive semigroup {Z(t)}t≥0

Z(t1)Z(t2) = Z(t1+ t2),t1,t2≥ 0.

If H is a (separable) Hilbert space corresponding to a particular scattering system and

{U(t)}t∈Ris a unitary group defined on H describing the evolution of the system, the basic

premises of the Lax-Phillips theory include the assumption of the existence of an incoming

subspace D− and an outgoing subspace D+ with respect to {U(t)}t∈R which are assumed

furthermore to be orthogonal to each other. Denoting by P−and P+respectively the projec-

tions on the orthogonal complements of D−and D+in H, and letting K = H ⊖ (D−⊕ D+),

the Lax-Phillips semigroup {Z(t)}t≥0defined by

Z(t) = P+U(t)P−= PKU(t)PK,t ≥ 0, (1)

annihilates D±and maps K into itself. The subspace K contains the scattering resonances

and the Lax-Phillips semigroup {Z(t)}t≥0describes their time evolution. In the Lax-Phillips

framework resonances are associated with pure states in the Hilbert space H.

A basic difficulty encountered in the work on application of the Lax-Phillips theory in

quantum mechanics originates from the fact that in this theory the continuous spectrum

of the generator of evolution is required to be unbounded from below as well as from above.

Hence a formalism utilizing the original structure of the theory, such as in Ref. 14, is not suit-

able for application to large classes of scattering problems in quantum mechanics (except for

limited types of problems, such as the Stark effect Hamiltonian17, or problems in a relativisti-

cally covariant framework15,16, which can be analyzed by direct mapping to the Lax-Phillips

structure. The case of a Schr¨ odinger equation with compactly supported potential may also

be analyzed within the Lax-Phillips framework through the use of the invariance principle of

wave operators18). The subject of the present paper is a theoretical framework, termed the

semigroup decomposition of resonance evolution, developed with the goal of overcoming such

difficulties. Proposed by one of the authors (Y.S.) of the present article19,20, this formalism

makes use of the Sz.-Nagy-Foias theory of contraction operators and contractive semigroups

on Hilbert space21which, from the mathematical point of view, is the fundamental theory

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underlying the Lax-Phillips construction through the notion of model operators for C·0class

semigroups (see Section 2).

The presentation of the semigroup decomposition formalism in Ref. 20 is based on the

following assumptions:

(i) We are considering a scattering system consisting of a “free” unperturbed Hamiltonian

H0and a perturbed Hamiltonian H, both defined on a Hilbert space H.

(ii) ess suppσac(H0) = ess suppσac(H) = R+. For simplicity it is assumed further that the

multiplicity of the a.c. spectrum is one.

(iii) The Møller wave operators Ω±≡ Ω±(H0,H) exist and are complete.

(iv) The S-matrix in the energy representation (the spectral representation for H0), denoted

by˜S(·) has an extension to a meromorphic function S(·) in an open, simply connected,

region Σ ⊂ C such that Σ∩R is an open interval in R. The operator valued function S(·)

is holomorphic in Σ ∩ C+and has a simple pole (we generalize to the case of multiple

poles in Section 3 below) at a point z = µ ∈ Σ ∩ C−and no other singularity in Σ, the

closure of Σ.

It is shown in Ref. 20 that there exists a dense set Λ ⊂ Hac(H) and a well defined state

ψµ∈ Hac(H) such that for any g ∈ Λ and any f ∈ Hacthe properties (i)-(iv) above induce,

for positive times, a decomposition of matrix elements of the evolution U(t) in the form

(g,U(t)f)Hac(H)= R(g,f;t) + α(g,µ)(ψµ,f)Hac(H)e−iµt,t ≥ 0.(2)

In a sense to be made precise in the next section the second term on the right hand side of

Eq. (2) originates from an evolution semigroup of Lax-Phillips type and the eigenvalue of the

generator of this semigroup is exactly µ, i.e., the point of singularity of the S-matrix. The

quantity R(g,f;t) on the right hand side of Eq. (2) is what we shall call a background term.

We note that if in Eq. (2) we choose f to be orthogonal to ψµthen the exponentially decaying

semigroup term (second term on the r.h.s. of Eq. (2) ) vanishes. We call ψµan approximate

resonance state and note that the characterization of ψµas an approximate resonance state

rather than as an exact resonance state stems from the fact that one can show (see Ref. 20)

that there is no choice of g and f that makes the backgound term R(g,f;t) vanish.

An explicit expression for the approximate resonance state ψµ is provided in Ref. 20.

It is shown there that, if we denote by {|E−?}E∈R+ the set of outgoing solutions of the

Lippmann-Schwinger equation (using Dirac’s notation), then ψµis given by

ψµ=

1

2πi

?

R+dE

1

E − µ|E−?.(3)

Following the introduction of approximate resonance states, the present paper discusses some

generalizations. Thus, in Section 3 we assume that the region Σ ∩ C−contains multiple

resonance poles of the S-matrix S(·), say at z = µ1,...,µn and obtain the form of the

expression for the approximate resonance states and semigroup decomposition of evolution

matrix elements in this case. In particular, we apply the semigroup decompostion to the

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survival amplitude, a central notion in the characterization of the time evolution of resonances.

Theorem 5 below then provides an a priori upper bound on the size of the background term

in this case.

As a final remark we note that a modification of the Lax-Phillips theory was recently used

by H. Baumgartel for the description of scattering resonances in certain quantum mechanical

problems22(see also Ref. 23). In particular, the assumption of orthogonality of D±, essential

in the context of the original Lax-Phillips formalism, is replaced in Ref. 22 by the require-

ment that an incoming subspace D−and an outgoing subspace D+exist and the respective

projections commute. The modified assumptions on D±, accompanied by certain assump-

tions on S-matrix analyticity properties, result in a modified Lax-Phillips structure which

is then applied to the Friedrichs model, leading to the construction of appropriate Gamow

type vectors24associated with scattering resonances. The framework presented in Ref. 22

has several points of intersection with the semigroup decomposition formalism discussed in

the present paper. The nature of these relationships will be discussed elsewhere.

The rest of the paper is organized as follows: In Section 2 we describe the formalism

providing the semigroup decomposition of resonance evolution starting with a short discussion

of the functional model for C·0continuous contractive semigroups followed by a description

of the semigroup decomposition formalism introduced in Ref. 19,20. In Section 3 we extend

the framework of Ref. 19,20 to the case of multiple resonances and, furthermore, find an

estimate on the size of the background term in the expression for the time evolution of the

survival probability of a resonance. In Section 4 we analyze a simple but illuminating example

involving a one dimensional model of scattering from a square barrier potential. Section 5

contains a short summary of the contents of the paper and some indication on further possible

courses of investigation.

2The semigroup decomposition for

resonance evolution

2.1Classification of contractive semigroups

Several distinct classes of contractive semigroups are identified within the framework of the

Sz.-Nagy-Foias theory. Let {T(t)}t≥0be a strongly contractive semigroup defined on a Hilbert

space H. The classes C0·, C·0, C1·, C·1are defined by

{T(t)}t∈R+ ∈ C0·

{T(t)}t∈R+ ∈ C·0

{T(t)}t∈R+ ∈ C1·

{T(t)}t∈R+ ∈ C·1

The classes Cαβwith α,β = 0,1 are then defined by

ifT(t)h → 0, ∀h ∈ H

T∗(t)h → 0, ∀h ∈ H

T(t)h ?→ 0, ∀h ∈ H, h ?= 0

T∗(t)h ?→ 0, ∀h ∈ H, h ?= 0

if

if

if

Cαβ= Cα·∩ C·β, α,β = 0,1.

The semigroup {Z(t)}t∈R+ describing the time evolution of resonances in the Lax-Phillips

theory is readily characterized by the fact that {Z∗(t)}t∈R+ belongs to the class C·0. The

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structure of the Lax-Phillips outgoing spectral (and translation) representation is then deter-

mined by that of the functional model21,25for C·0class semigroups provided by the Sz.-Nagy-

Foias theory. We say an operator A is a model operator25for a given class C of operators

if every operator in C is similar to a multiple of a part of A (a part of an operator A is a

restriction of A to one of its invariant subspaces). By a functional model we mean that the

model operator for a given class C has a canonical representation on suitable function spaces.

For a C·0class semigroup {T(t)}t≥0the associated functional model is essentially obtained

through a procedure of isometric dilation of the cogenerator of {T(t)}t≥0and the similarity

mapping to the functional model is in fact a unitary transformation.

2.2The functional model for C·0semigroups

We turn now to a brief description of the functional model for semigroups in the class C·0

. Denote by C+the upper half of the complex plane and let H2

of vector valued functions analytic in the upper half-plane and taking values in a separable

Hilbert space N. The set of boundary values on R of functions in H2

by H2

of N valued functions analytic in the lower half-plane is denoted by H2

is the isomorphic Hilbert space consisting of boundary values on R of functions in H2

Define {u(t)}t∈R, a family of unitary, multiplicative operators u(t) : L2

[u(t)f](σ) = e−iσtf(σ),

N(C+) be the Hardy space

N(C+), denoted below

N+(R), is a Hilbert space isomorphic to H2

N(C+). In a similar manner the Hardy space

N(C−) and H2

N−(R)

N(C−).

N(R) ?→ L2

N(R) by

f ∈ L2

N(R), σ ∈ R. (4)

Assume that {T(t)}t≥0 is a C·0 class semigroup defined on a Hilbert space K.

semigroup {ˆT(t)}t≥0, defined on a Hilbert spaceˆK, be the functional model for {T(t)}t≥0and

let W : K ?→ˆK be the similarity transforming {T(t)}t≥0into its functional model {ˆT(t)}t≥0

i.e.,ˆT(t) = WT(t)W−1. Then there exists a Hilbert space N such thatˆK is a closed subspace

of H2

N+(R), W is unitary, and the functional model is given by

Let the

ˆT(t) = WT(t)W∗= PˆKu∗(t)|ˆK,t ≥ 0.(5)

Here PˆKis the orthogonal projection from H2

N+(R) ontoˆK, the subspaceˆK is given by

ˆK = H2

N+(R) ⊖ ΘT(·)H2

N+(R),(6)

and ΘT(·) : H2

course, on {T(t)}t≥0) i.e., an operator valued function with the properties:

1. For each σ ∈ R the operator ΘT(σ) : N ?→ N is the boundary value at σ of an operator

valued function ΘT(·) analytic in the upper half-plane.

2. ?ΘT(z)?N≤ 1 for Imz > 0.

3. ΘT(σ), σ ∈ R is, pointwise, a unitary operator on N.

N+(R) ?→ H2

N+(R) is an inner function26,29,30for H2

N+(R) (depending, of

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