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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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The operator valued function ΘT(·) is, in fact, the characteristic function21of the cogenerator
of the semigroup {ˆT(t)}t≥0(or {T(t)}t≥0).
Let P+be the orthogonal projection of L2
symbol u(t) (see, for example, Ref.26,27 and references therein), is an operator Tu(t):
H2
N+(R) defined by
N(R) on H2
N+(R). The Toeplitz operator with
N+(R) ?→ H2
Tu(t)f
def
= P+u(t)f,f ∈ H2
N+(R). (7)
We note that {Tu(t)}t≥0is a strongly contractive semigroup on H2
Ref. 1,19,21,28. Taking the conjugate ofˆT(t) in H2
N+(R) (see, for example,
N+(R) and using Eq. (5) one finds that
ˆT∗(t) = WT∗(t)W∗= Tu(t)|ˆK,t ≥ 0.(8)
It follows from the discussion above that the Lax-Phillips semigroup {Z(t)}t≥0 has a
functional model in the form of Eq. (8) (recall that {Z∗(t)}t≥0is a C·0class semigroup), i.e.,
if we denote the functional model for {Z(t)}t≥0by {ˆZ(t)}t≥0then we have
ˆZ(t) = WZ(t)W∗= Tu(t)|ˆK,
whereˆK ⊂ H2
ˆK = H2
t ≥ 0 (9)
N+(R) is an invariant subspace for {Tu(t)}t≥0given by
N+(R) ⊖ ΘZ(·)H2
N+(R) (10)
and the inner function ΘZ(·) and the Hilbert space N are determined by {Z(t)}t≥0.
semigroup {ˆZ(t)}t≥0of the form given by Eq. (9) and Eq. (10) is referred to in Ref. 20 as a
Lax-Phillips type semigroup.
A central theorem of the Lax-Phillips theory, corresponding to an important result in the
Sz.-Nagy-Foias theory relating the spectrum of a completely non-unitary (cnu) contraction to
points of singularity of the characteristic function states the following
A
Theorem 1 Denote byˆB the generator of a Lax-Phillips type semigroup {ˆZ(t)}t≥0. If Imµ <
0, then µ belongs to the point spectrum ofˆB if and only if Θ∗
Z(µ) has a nontrivial null space.
We note that the analytic continuation of ΘZ(z) to the lower half-plane is given by
ΘZ(z)
def
= (Θ∗
Z(z))−1, Imz < 0
and so a null space for Θ∗
of the Lax-Phillips theory the characteristic function ΘZ(·) for the Lax-Phillips semigroup is
identical to the Lax-Phillips S-matrix and its poles are the scattering resonances. As will be
seen below, the situation is a bit more involved in the semigroup decomposition formalism.
We do not elaborate here further on the relations between the functional model for C·0
semigroups discussed above and the full structure of the Lax-Phillips spectral representations
and wave operators. The reader is referred to Ref. 1,21.
Z(µ) implies the existence of a pole for ΘZ(z) at z = µ. In the case
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2.3The semigroup decomposition
In order to apply the functional model for C·0 semigroups , which is at the heart of the
Lax-Phillips structure, to the description of resonance evolution it is necessary to relate, for
t ≥ 0, the evolution U(t) defined on the Hilbert space H of the scattering problem to the
Toeplitz evolution semigroup Tu(t)of Eq. (7) defined on H2
according to Eq. (9), to a subspaceˆK of H2
function Θˆ Z(·). In the framework of the Lax-Phillips theory this relation is guaranteed by
the special properties of the Lax-Phillips incoming and outgoing subspaces D±(with D−and
D+denoting, respectively, the incoming and outgoing subspace), since in this case the Lax-
Phillips semigroup is a C·0semigroup. However, for many quantum mechanical scattering
problems one usually cannot find subspaces with the properties of D±. A way of overcoming
this difficulty, proposed in Ref.19 is to combine the standard functional model for C·0
semigroups with the notion of a quasi-affine mapping (see, for example, Ref. 21, Pg. 70):
N+(R) and then restrict the latter,
N+(R) associated with an appropriate inner
Definition 1 (Quasi-affine mapping) A quasi-affine map from a Hilbert space H1into a
Hilbert space H0is a linear, one to one continuous mapping of H1into a dense linear manifold
in H0. If A ∈ B(H1) and B ∈ B(H0) then A is a quasi-affine transform of B if there is a
quasi-affine map θ : H1?→ H0such that θA = Bθ.
The following theorem is proved in Ref. 19 for a scattering system consisting of unperturbed
and perturbed Hamiltonians, respectively H0and H, having semibounded continuous spec-
trum:
Theorem 2 (Outgoing/Incoming contractive nesting) Let H0 and H be self-adjoint
operators on a Hilbert space H. Let {U(t)}t∈Rbe the unitary evolution group on H generated
by H [i.e, U(t) = exp(−iHt)]. Denote by Hac(H0) and Hac(H), respectively, the absolutely
continuous subspaces of H0and H. Assume that the absolutely continuous spectrum of H0
and H has multiplicity one and that essSuppσac(H0) = essSuppσac(H) = R+. Assume
furthermore that the Møller wave operators Ω±≡ Ω±(H0,H) : Hac(H0) ?→ Hac(H) exist and
are complete. Then there are mappingsˆΩ±: Hac(H) ?→ H2
(i)ˆΩ±are contractive quasi-affine mappings of Hac(H) into H2
(ii) For every t ≥ 0 the evolution U(t) is a quasi-affine transform of the Toeplitz operator
Tu(t)via the mappingˆΩ±i.e., for every f ∈ Hac(H) we have
ˆΩ±U(t)f = Tu(t)ˆΩ±f
+(R) such that
+(R).
t ≥ 0.(11)
?
We call the triplet (Hac(H),H2
H2
contractive nesting of Hac(H) into H2
Define
+(R),ˆΩ−) the incoming contractive nesting of Hac(H) into
+(R) and denote fin=ˆΩ−f. Similarly, the triplet (Hac(H),H2
+(R) and we denote fout=ˆΩ+f.
+(R),ˆΩ+) is the outgoing
ΞˆΩ+
def
=ˆΩ∗
+H2
+(R).
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Then, sinceˆΩ∗
sinceˆΩ∗
We note that in Ref. 20 a dense set ΛˆΩ+, analogous to ΞˆΩ+, is defined somewhat differently,
i.e., ΛˆΩ+
unlike that of ΛˆΩ+, allows for a full characterization of approximate resonance states. Using
Theorem 2 we have, for every g ∈ ΞˆΩ+and f ∈ Hac(H) and for t ≥ 0
+is quasi-affine, the linear space ΞˆΩ+⊂ Hac(H) is dense in Hac(H). Moreover,
+is one to one, for each g ∈ ΞˆΩ+there is a unique ˜ g ∈ H2
+(R) such that g =ˆΩ∗
+˜ g.
def
=ˆΩ∗
+ˆΩ+Hac(H). However, it will be seen below that the definition of ΞˆΩ+above,
(g,U(t)f)Hac(H)= (ˆΩ∗
+˜ g,U(t)f)Hac(H)=
= (˜ g,Tu(t)ˆΩ+f)H2
+(R)= (˜ g,Tu(t)fout)H2
+(R),t ≥ 0(12)
Following the definitions of the incoming and outgoing nestings of Hac(H) into H2
is natural to define the nested S-matrix
+(R) it
Snest
def
=ˆΩ+ˆΩ−1
−.
Let U : Hac(H0) ?→ L2(R+) be the unitary transformation of Hac(H0) onto the spectral
representation for H0(also called the energy representation for H0). If S = (Ω−)∗Ω+is the
scattering operator associated with H0and H then˜S(·) : L2(R+) ?→ L2(R+) defined by
˜S(·)def
is the energy representation of the S-matrix. Let PR+ : L2(R) ?→ L2(R) be the orthogonal
projection in L2(R) on the subspace of functions supported on R+and define the inclusion
map I : L2(R+) ?→ L2(R) by
?
0,
= USU∗
(13)
(If)(σ) =
f(σ),σ ≥ 0
σ < 0.
(14)
Then the inverse I−1: PR+L2(R) ?→ L2(R+) is, of course, one to one on PR+L2(R). Let
θ : H2
+(R) ?→ L2(R+) be a map given by
θf = I−1PR+f,f ∈ H2
+(R).(15)
By a theorem of Van Winter31, θ is a quasi-affine transform mapping H2
The adjoint map θ∗: L2(R+) ?→ H2
explicit expression for θ∗is provided by the following lemma19:
+(R) into L2(R+).
+(R) is then also a contractive quasi-affine map. An
Lemma 1 Let I : L2(R+) ?→ L2(R) be the inclusion map defined in Eq. (14). Let P+be the
orthogonal projection of L2(R) onto H2
+(R). Then for every f ∈ L2(R+) we have
θ∗f = P+If,f ∈ L2(R+).(16)
?
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It is shown in Ref. 19 that the nested S-matrix can be expressed in the form
Snest= θ∗˜S(·)(θ∗)−1. (17)
Following Ref. 20 we now use assumption (iv) in Section 1. The S-matrix˜S(·) is then the
restriction of its extension S(·) on R+. Under these assumptions S(·) has, in the region Σ, a
representation of the form (see Ref. 20)
S(z) = Bµ(z)S′(z),z ∈ Σ
where
Bµ(z) =z − µ
z − µ,
z ∈ C\{µ}
(18)
and S′(·) is analytic and has no zeros in Σ. Restricting S(·) to the positive real axis we obtain
˜S(E) =˜Bµ(E)˜S′(E),E ≥ 0(19)
where by definition˜Bµ(E)
and˜Bµ(·) are considered here as multiplicative unitary operators on L2(R+) (moreover, they
are pointwise unitary a.e. for E ≥ 0). Moreover, Bµ(·) can be regarded as a multiplicative
operator on L2(R). In fact, considered as a multiplicative operator on H2
is a Blaschke factor (the definition of Blaschke products and Blaschke factors can be found,
for example in Ref. 29,30, see e.g., Eq. (29) below). Such a factor is the simplest example
of an inner function for H2
+(R). We make use of this fact through the following proposition,
not stated as such, but implicitly used in Ref. 20:
def
= Bµ(E) and˜S′(E)
def
= S′(E) for E ≥ 0. We note that both˜S′(·)
+(R) ⊂ L2(R), Bµ
Proposition 1 Let˜Bµ(·) : L2(R+) ?→ L2(R+) be defined by˜Bµ(E) = Bµ(E), E ≥ 0 where
Bµ(·) is defined in Eq. (18). Let θ∗: L2(R+) ?→ H2
in Eq. (15). LetˆKµ⊂ H2
+(R) be the adjoint of the map θ defined
−(R) be subspaces defined by
+(R) andˆKµ⊂ H2
ˆKµ
def
= H2
+(R) ⊖ Bµ(·)H2
+(R),
ˆKµ
def
= H2
−(R) ⊖ Bµ(·)H2
−(R),
where Bµ(z) = (z − µ)(z − µ)−1and denote by PˆKµand PˆKµthe orthogonal projections of
L2(R) onˆKµandˆKµrespectively. For every f ∈ L2(R+) we then have
θ∗˜Bµf = Bµθ∗f + PˆKµBµPˆKµθ∗f .
here θ∗: L2(R+) ?→ H2
projection of L2(R) onto H2
−(R).
(20)
−(R) and θ∗f = P−If with I defined in Eq. (14) and P−the orthogonal
?
Proof: Using Eq. (16) in Lemma 1 we get
θ∗˜Bµf = P+I˜Bµf = P+BµIf = P+Bµ(P++ P−)If = P+Bµθ∗f + P+BµP−θ∗f
Eq. (20) then follows from the fact, proved in Ref. 20, that P+BµP−= PˆKµBµPˆKµand from
the property of Bµ(·) of being an inner function for H2
+(R).
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?
We note that since Bµ(·) is an inner function Eq. (9), (10) and Theorem 1 imply that
Tu(t)PˆKµ=ˆZ(t)PˆKµ= e−iµtPˆKµ,t ≥ 0. (21)
Combining Eq. (17), Eq. (19) and Eq. (20) we obtain
fout= Snestfin= θ∗˜S(θ∗)−1fin= θ∗˜Bµ˜S′(θ∗)−1fin=
= Bµθ∗˜S′(θ∗)−1fin+ PˆKµBµPˆKµθ∗˜S′(θ∗)−1fin.(22)
Using the decomposition of foutfrom Eq. (22) in the r.h.s. of Eq. (12) and applying Eq. (21)
we obtain the semigroup decomposition for t ≥ 0 of the time evolution corrsponding to the
resonance at z = µ
(g,U(t)f)Hac(H)= (˜ g,Tu(t)fout)H2
+(R)=
= (˜ g,u(t)Bµθ∗˜S′(θ∗)−1fin)H2
+(R)+ e−iµt(˜ g,Bµθ∗˜S′(θ∗)−1fin)H2
+(R). (23)
As is seen above in Eq. (21), the exponential decay in the second term on the r.h.s of Eq. (23)
originates with the semigroupˆZ(t). The first term on the r.h.s. of Eq. (23) is the background
term and is responsible for deviations from a purely exponential decay law.
3Approximate resonance states
It is an interesting fact that the semigroup decomposition described in the previous section
associates a unique state in Hac(H) with a resonance pole at z = µ (Imµ < 0). The following
theorem is proved in Ref. 20
Theorem 3 (approximate resonance state) Under the assumptions of Theorem 2, let
˜S : L2(R+) ?→ L2(R+) be the S-matrix in the energy representation defined in Eq. (13).
Assume that˜S(·) is the restriction to R+of a function S(·) meromorphic in an open region
Σ with a single, simple pole at a point z = µ, µ ∈ Σ ∩ C−. For any f ∈ Hac(H) define
fout=ˆΩ+f and fin=ˆΩ−f. There exists a unique state ψµ∈ Hac(H) such that
fout= Bµθ∗˜S′(θ∗)−1fin+|Imµ|
π
(ψµ,f)Hac(H)xµ
(24)
where θ∗is the map given by lemma 1, Bµis given in Eq. (18),˜S′is defined by Eq. (19) and
xµ∈ H2
Combining Eq. (24) and Eq. (23) we can write the semigroup decomposition in the form
+(R) is given by xµ(σ) = (σ − µ)−1, σ ∈ R.
?
(g,U(t)f)Hac(H)= (˜ g,u(t)Bµθ∗˜S′(θ∗)−1fin)H2
+(R)
+Imµ
π
e−iµt(˜ g,xµ)H2
+(R)(ψµ,f)Hac(H),t ≥ 0(25)
10
Page 11
where g ∈ ΞˆΩ+and ˜ g = (ˆΩ∗
the exponential decay of the second term on the r.h.s. of Eq. (25) is called below the Hardy
space resonance state. The state ψµ∈ Hac(H) whose existence is implied by Theorem 3 is
called approximate resonance state. We observe that if in Eq. (25) we choose f ∈ Hac(H)
orthogonal to ψµthen the second term on the r.h.s. of that equation is identically zero.
Denote by {|E−?}E∈R+ the set of outgoing solutions of the Lippmann-Schwinger equation.
For every f ∈ Hac(H) we have
(U(Ω−)∗f)(E) = ?E−|f?,
+)−1g. The eigenstate xµ∈ H2
+(R) of the semigroupˆZ(t) providing
E ∈ R+
(26)
It is shown in Ref. 20 that an explicit expression for the approximate resonance state ψµis
given by
1
2πi
ψµ=
?
R+dE
1
E − µ|E−?. (27)
In this section we explore several properties of approximate resonance states ψµ. Our first
step is to extend the discussion above to the case of multiple resonances:
Theorem 4 (multiple resonance case) Under the assumptions of Theorem 2, let˜S : L2(R+) ?→
L2(R+) be the S-matrix in the energy representation defined in Eq. (13). Assume that˜S(·) is
the restriction to R+of a function S(·) meromorphic in the open region Σ with n simple poles
at points z = µi, i = 1,...,n, µi∈ Σ ∩ C−. Then there exist n distinct states {ψΣ
ψΣ
µi∈ Hac(H), such that for every f ∈ Hac(H) we have
µi}i=1,...,n,
fout= Bµ1...µnθ∗˜S′(θ∗)−1fin+
n
?
j=1
|Imµj|
π
n
?
i?=j
i=1
µj− µi
µj− µi(ψΣ
µj,f)Hac(H)xµj
(28)
where
Bµ1...µn(z)def
=
n
?
i=1
z − µi
z − µi.(29)
In Eq. (28)˜S′(·) is the restriction to R+of a function S′(·) analytic in Σ and having no poles
in Σ and xµj(σ) = (σ − µj)−1, σ ∈ R. The states ψΣ
µj, j = 1,...,n are given by
ψΣ
µj=
?
R+dE
n
?
i?=j
i=1
E − µi
E − µi
1
E − µj|E−?. (30)
?
Proof: Assume that S(·), the extension of˜S(·) from R+into Σ∪R+has n simple poles in
Σ∩C−. Then, applying the same arguments as in Ref. 20, we find that S(·) can be factorized
in Σ in the form
S(z) = Bµ1...µn(z)S′(z)
11
Page 12
where Bµ1...µn, defined in Eq. (29), is a finite Blaschke product and S′(·) has no poles in Σ.
In addition we have, of course
˜S(E) =˜Bµ1...µn(E)˜S′(E),E ≥ 0.
The semigroup decomposition then follows exactly as in Section 1 with Bµ1...µnand˜Bµ1...µn
replacing Bµand˜Bµrespectively. For the resonance term in Eq. (22) we get in this case
P+Bµ1...µnP−θ∗˜S′(θ∗)−1fin,f ∈ Hac(H)
Recalling that
(P+f)(σ) =
1
2πi
?∞
−∞
dσ′
1
σ − σ′+ i0f(σ′),f ∈ L2(R), σ ∈ R
and
(θ∗f)(σ) =
1
2πi
?∞
0
dE
1
E − σ + i0f(E),f ∈ L2(R+)
(see Ref. 19) we obtain
(P+Bµ1...µnP−θ∗˜S′(θ∗)−1fin)(σ) =
=−1
4π2
0
?∞
n
?
dE
?∞
−∞
dσ′
1
σ − σ′+ i0
?∞
n
?
n
?
i?=j
?∞
i=1
σ′− µi
σ′− µi
µj− µi
µj− µi
1
E − σ′+ i0
1
E − µj
˜S′(E)((θ∗)−1fin)(E) =
=
j=1
|Imµj|
π
1
σ − µj
0
dE
i=1
˜S′(E)((θ∗)−1fin)(E) =
=
n
?
j=1
|Imµj|
π
xµj(σ)
n
?
i?=j
i=1
µj− µi
µj− µi
0
dE
1
E − µj
n
?
i?=j
i=1
E − µi
E − µi
˜S(E)((θ∗)−1fin)(E) =
=
n
?
j=1
|Imµj|
π
xµj(σ)
n
?
i?=j
i=1
µj− µi
µj− µi
?∞
0
dE
1
E − µj
n
?
i?=j
i=1
E − µi
E − µi
(U(Ω−)∗f)(E) =
=
n
?
j=1
|Imµj|
π
xµj(σ)
n
?
i?=j
i=1
µj− µi
µj− µi
?∞
0
dE
1
E − µj
n
?
i?=j
i=1
E − µi
E − µi
?E−|f?
where xµj∈ H2
(σ − µj)−1. Defining the states ψΣ
+(R) is the Hardy space resonance state corresponding to µj i.e., xµj(σ) =
µj, j = 1,...n according to Eq. (30) we obtain
(P+Bµ1...µnP−θ∗˜S′(θ∗)−1fin)(σ) =
n
?
j=1
|Imµj|
π
n
?
i?=j
i=1
µj− µi
µj− µi(ψΣ
µj,f)Hac(H)xµj(σ)(31)
12
Page 13
This proves Theorem 4.
?
We observe that Eq. (30) is a generalization of Eq. (27). Hence ψΣ
resonance state corresponding to the pole of S(·) at z = µj. Combining Eq. (31) and Eq.
(23) we get the semigroup decomposition for the multi-resonance case
µjis the approximate
(g,U(t)f)Hac(H)= (˜ g,u(t)Bµ1...µnθ∗˜S′(θ∗)−1fin)H2
n
?
+(R)
+
j=1
|Imµj|
π
n
?
i?=j
i=1
µj− µi
µj− µi(ψΣ
µj,f)Hac(H)(˜ g,xµj)H2
+(R)e−iµjt,t ≥ 0. (32)
The approximate resonance states in Eq. (30) and semigroup decomposition of Eq. (28)
and Eq. (32) depend, of course, on the region Σ. If {µj}j=1,...,nare the poles in Σ∩C−of the
meromorphic extension S(·) of the S-matrix˜S(·), the approximate resonance state defined
in Eq. (30) for a resonance at z = µj is therefore denoted by ψΣ
arguments the exact form of Σ is irrelevant and it is useful to define the notion of an n’th
order approximate resonance state:
µj. However, for certain
Definition 2 (n’th order approximate resonance state) If the number of poles of S(·)
entering into the definition of the approximate resonance state ψΣ
µjitself, is n we say that ψΣ
µjis an n’th order approximate resonance state for the resonance
at z = µj. In particular, regardless of the exact nature of the region Σ, the zero’th order
approximate resonance state is always defined to be given by Eq. (27) with µ = µj and is
denoted by ψ(0)
µj.
µjin Eq. (30), not including
Remark: Note that in general there are many choices of the n resonance poles (different
than µj) included in the construction of what we call an n’th order approximation ψ(n)
cases that the nature of the region Σ is irrelevant and only the order of the approximate
resonance state is significant we replace the notation ψΣ
approximate resonance state considered.
µj. In
µjby ψ(n)
µj, where n is the order of the
The semigroup decomposition and approximate resonance states for the multi–resonance
case possess some interesting properties. For example, we have
(ψΣ
µj,ψΣ
µk)Hac(H)= (ψ(0)
µj,ψ(0)
µk)Hac(H)=
?
R+dE
1
E − µj
1
E − µk
and, in particular
?ψΣ
µj?2
Hac(H)= ?ψ(n)
µj?2
Hac(H)= ?ψ(0)
µj?2
Hac(H)=
?
R+dE
1
|E − µj|2. (33)
We see that, although the definition of ψΣ
Σ ∩ C−, the scalar product of ψΣ
be extended to a meromorphic function in a region Σ′⊃ Σ (we keep the notation S(·) for
the extended function) and S(·) has now m > n simple poles in Σ′∩ C−we may calculate
µiin Eq. (30) depends on all of the poles {µj}j=1,...,n⊂
µiand ψΣ
µjdepends only on µiand µj. In fact, if S(·) can
13
Page 14
approximate resonance states of order m − 1 for all resonances in Σ′according to Eq. (30).
However, for µj,µk∈ Σ we would still have (ψΣ′
products (and norms) are independent of the order of the approximate states when we enlarge
the region Σ. In particular we have ?ψΣ
µj.
An interesting question is whether the peculiar properties of scalar products and norms
of the approximate resonance states mentioned above characterize also the time evolution of
these states. We shall see below that, at least partially, the answer to this question is positive.
For this we consider one of the basic notions associated with resonance evolution, i.e., that of
the survival amplitude
(ψΣ
(ψΣ
µj,ψΣ
µj,ψΣ′
µk)Hac(H)= (ψΣ
µj,ψΣ
µk)Hac(H)i.e., scalar
µj?Hac(H)= ?ψ(0)
µj?Hac(H)for every region Σ containing
AψΣ
µj(t)def
=
µj,U(t)ψΣ
µj)Hac(H)
µj)Hac(H)
. (34)
Making use of Eq. (30) and (33) we get a simple expression for this quantity
Aψ(0)
µj(t) = ?ψ(0)
µj?−2
Hac(H)
?
R+
1
|E − µj|2e−iEt,t ≥ 0
where ?ψ(0)
for the approximate resonance state ψΣ
form as for a single resonance. This suggests that the semigroup decompostion of the survival
amplitude for the multiple resonance case is similar to that of a single resonance. When
combined with an important characterization of approximate resonance states in the form of
Lemma 2 below, such considerations lead to the following useful a priori estimate on the size
of the background term in the semigroup decomposition of the survival amplitude:
µj? is given in Eq. (33). Again, we see that the expression for the survival amplitude
µjdepends only on the pole at z = µjand has the same
Theorem 5 Let AψΣ
the background term Rµj(t) be defined by the relation
µj(t), j = 1,...,n be the survival amplitude defined in Eq. (34) and let
AψΣ
µj(t) = Rµj(t) + e−iµjt,t ≥ 0. (35)
Then we have
|Rµj(t)| ≤
?xµj?4
?ψ(0)
H2
+(R)
µj?4
Hac(H)
− 1
1/2
,t ≥ 0. (36)
where xµj(σ) = (σ−µj)−1is the Hardy space resonance state and ψ(0)
order approximate resonance state corresponding to the resonance at z = µj.
µj∈ Hac(H) is the zero’th
?
Proof: We first have
Proposition 2 For j = 1,...,n, let ψΣ
µjbe defined by Eq. (30) and let AψΣ
µj(t) be the survival
amplitude defined in Eq. (34). Then
AψΣ
µj(t) = ?ψ(0)
µj?−2
Hac(H)(xµj,u(t)Bµjθ∗˜S′
µj(θ∗)−1ψ(0)
µj,in)H2
+(R)+ e−iµjt,t ≥ 0. (37)
14
Page 15
where xµj∈ H2
approximate resonance state corresponding to the pole at µj, and˜S′
(19), i.e.,
˜S′
+(R) is the Hardy space resonance state and ψ(0)
µj∈ Hac(H) is the zero’th order
µj(·) is defined as in Eq.
µj(E) =E − µj
E − µj
˜S(E) =˜Bµj(E)˜S(E).
?
Proof of Proposition 2: We need first the following easily proved, but important,
lemma
Lemma 2 For j = 1,...,n, let ψΣ
µjbe defined by Eq. (30). Define
Bµ1...ˆ µk...µn(z)def
=
n
?
i?=k
i=1
z − µi
z − µi
where {µj}j=1,...,nare the poles of S(·) in Σ, and let xµj(σ) = (σ − µj)−1. Then we have
ψΣ
+Bµ1...ˆ µj...µnxµj.
In particular ψ(0)
+xµj.
Proof of Lemma 2: It is proved in Ref. 19 that, if Ω±are the Møller wave operators,
U : Hac(H0) ?→ L2(R+) the mapping to the energy representation for H0 (see Eq. (13)
above) and θ∗the map given in Lemma 1, then the quasi-affine nesting mapsˆΩ±are given by
ˆΩ±= θ∗U(Ω∓)∗, hence we haveˆΩ∗
µj=ˆΩ∗
µj=ˆΩ∗
?
+= Ω−U∗θ. Furthermore, by the definition of θ we have
(θBµ1...ˆ µj...µnxµj)(E) =
n
?
i?=j
i=1
E − µi
E − µi
1
E − µj,E ∈ R+.(38)
Moreover, according to Eq. (26) for every g ∈ L2(R+) we have
Ω−U∗g =
?
R+dE |E−?g(E). (39)
Applying Eq. (39) with g = θBµ1...ˆ µj...µnxµjand comparing with Eq. (30) proves the lemma
?
Note that by Lemma 2 we have (ˆΩ∗
+)−1ψΣ
µj= Bµ1...ˆ µj...µnxµj. Hence by Eq. (32) we get
AψΣ
µj(t) = ?ψΣ
µj?−2
= ?ψΣ
|Imµk|
Hac(H)(ψΣ
µj,U(t)ψΣ
µj)Hac(H)=
µj?−2
Hac(H)(Bµ1...ˆ µj...µnxµj,u(t)Bµ1...µnθ∗˜S′(θ∗)−1ψΣ
n
?
i?=k
µj,in)H2
+(R)
+
n
?
k=1
π
i=1
µk− µi
µk− µi
(ψΣ
µk,ψΣ
?ψΣ
µj)Hac(H)
µj?2
Hac(H)
(Bµ1...ˆ µj...µnxµj,xµk)H2
+(R)e−iµkt
(40)
15
Page 16
In the second term on the r.h.s. of Eq. (40) we first separate the term with k = j and get
n
?
k=1
|Imµk|
π
n
?
i?=k
i=1
µk− µi
µk− µi
(ψΣ
µk,ψΣ
?ψΣ
µj)Hac(H)
µj?2
Hac(H)
(Bµ1...ˆ µj...µnxµj,xµk)H2
+(R)e−iµkt=
=
n
?
k?=j
k=1
|Imµk|
π
n
?
i?=k
i=1
µk− µi
µk− µi
(ψΣ
µk,ψΣ
?ψΣ
µj)Hac(H)
µj?2
Hac(H)
(Bµ1...ˆ µj...µnxµj,xµk)H2
+(R)e−iµkt+ e−iµjt
,t ≥ 0
Here, use has been made of Eq. (42) below. The above expression can be further simplified
since xµk∈ˆKµk= H2
(Bµ1...ˆ µj...µnxµj,xµk)H2
+(R) ⊖ Bµk(·)H2
+(R) implies that for k ?= j we have
+(R)= (BµkBµ1...ˆ µj...ˆ µk...µnxµj,xµk)H2
+(R)= 0
where
Bµ1...ˆ µj...ˆ µk...µn(z)def
=
n
?
i?=k,j
i=1
z − µi
z − µi
.
The first term on the r.h.s. of Eq. (40) can also be simplified. We have
(Bµ1...ˆ µj...µnxµj,u(t)Bµ1...µnθ∗˜S′(θ∗)−1ψΣ
= (u(−t)Bµ1...ˆ µj...µnxµj,Bµ1...µnθ∗˜S′(θ∗)−1ψΣ
= (Bµ1...ˆ µj...µnu(−t)xµj,Bµ1...µnθ∗˜S′(θ∗)−1ψΣ
µj,in)H2
+(R)=
µj,in)H2
+(R)=
µj,in)H2
+(R)=
= (xµj,u(t)Bµjθ∗˜S′(θ∗)−1ψΣ
µj,in)H2
+(R),t ≥ 0
Moreover, using Lemma 2 we find that
θ∗˜S′(θ∗)−1ψΣ
µj,in= θ∗˜S′(θ∗)−1ˆΩ−ˆΩ∗
= θ∗˜S′(θ∗)−1θ∗˜S∗θBµ1...ˆ µj...µnxµj= θ∗˜Bµ1...µnθBµ1...ˆ µj...µnxµj=
= θ∗˜Bµjθxµj= θ∗˜Bµj˜S(θ∗)−1ˆΩ−ˆΩ∗
+Bµ1...ˆ µj...µnxµj=
+xµj= θ∗˜S′
µj(θ∗)−1ψ0
µj,in.
Recalling that Eq. (33) implies that ?ψΣ
is complete.
µj?Hac(H)= ?ψ(0)
µj?Hac(H)the proof of Proposition 2
?
From Proposition 2 we see that, independent of the region Σ, the semigroup decomposition
of the survival amplitude depends only on the zero’th order approximate resonance state.
Comparison of Eq. (37) and Eq. (35) gives
Rµj(t) = ?ψ(0)
µj?−2
Hac(H)(xµj,u(t)Bµjθ∗˜S′
µj(θ∗)−1ψ(0)
µj,in)H2
+(R),t ≥ 0,(41)
16
Page 17
This expression for Rµj(t) is identical to the zero’th order background term we would get
from Eq. (25) with f = g = ψ(0)
µj. We now exploit this fact to obtain the desired estimate in
Theorem 5. Applying Theorem 3 to the zero’th order approximate resonance state ψ(0)
obtain
ψ(0)
µjwe
µj,out=ˆΩ+ψ(0)
µj= Bµjθ∗˜S′
µj(θ∗)−1ψ(0)
µj,in+|Imµj|
π
?ψ(0)
µj?2
Hac(H)xµj.
Now, since bothˆΩ+andˆΩ∗
?ˆΩ+ˆΩ∗
+are contractive we note that Lemma 2 implies that ?ψ(0)
+(R)≤ ?xµj?H2
µj,out?H2
+(R). Therefore,
+(R)=
+xµj?H2
+(R). In addition in H2
+(R) we have xµj⊥ BµjH2
?Bµjθ∗˜S′
µj(θ∗)−1ψ(0)
µj,in?2
H2
+(R)+|Imµj|2
π2
?ψ(0)
µj?4
Hac(H)?xµj?2
H2
+(R)≤ ?xµj?2
H2
+(R).
It is easy to verify that
?xµj?2
H2
+(R)=
?∞
−∞
1
|σ − µj|2dσ =
π
|Imµj|,(42)
hence the inequality above can be written in the form
?Bµjθ∗˜S′
µj(θ∗)−1ψ(0)
µj,in?2
H2
+(R)≤
?
?xµ?4
H2
+(R)− ?ψ(0)
µj?4
Hac(H)
?
?xµj?−2
H2
+(R).(43)
Applying the Schwartz inequality to the r.h.s. of Eq. (41) and using the bound from Eq. (43)
we get the estimate in Eq. (36).
?
As mentioned above the background term cannot be identically zero. Hence deviations
from exponential decay of the survival probability are to be expected. In fact, it is easy to
verify that the survival probability behaves for short times as |Aψµj(t)|2= 1 − O(t2). Note
that Eq. (35) implies that at t = 0 we must have Rµj(0) = 0. This is also seen from Eq. (41),
since for t = 0 we have u(0) = 1 and xµj⊥ Bµj(·)H2
are then due to the fact that xµj?⊥ u(t)BµjH2
+(R). Deviations from exponential decay
+(R) for t > 0.
4 Example: Scattering from square barrier potential
In this section we apply the results of the previous two sections to a simple one dimensional
model with a square barrier potential. Although simple, this model provides a good illustra-
tion for the various results obtained above. In particular, we present numerical calculations of
approximate resonance states of various orders accompanied with plots of the time evolution
of the corresponding survival amplitudes and estimates of the size of the background term
following from Theorem 5.
The model we consider is a Schr¨ odinger equation in one spatial dimension on the half-
line R+with a square barrier potential. Thus we consider the free Hamiltonian H0= −∂2
acting on L2(R+) (where H0is defined as a self-adjoint extension to L2(R+) from the original
x
17
Page 18
domain of definition D(−∂2
given by H = H0+ V where V is a multiplicative operator (Vf)(x) = V (x)f(x) with
x) = {φ(x) ∈ W2
2(R+) | φ(x) = 0}) and the full Hamiltonian is
V (x) =
0,
V0,
0,
0 < x < a
a ≤ x ≤ b
b < x,
where b > a > 0 and we take V0> 0. In this case there are no bound state solutions of the
eigenvalue problem for H and we have σ(H) = σac(H) = R+. In order to find the scattering
states, calculate the S-matrix and finally the approximate resonance states for this problem
one solves the eigenvalue problem
−∂2
xψE(x) + V (x)ψE(x) = EψE(x),E ∈ R+
for the continuous spectrum generalized eigenfunctions ψE(x). Imposing boundary conditions
one finds that
coefficients in Eq. (44) are given by32
ψE(x) =
α1(k)sinkx,
α2(k)eik′x+ β2(k)e−ik′x,
α3(k)eikx+ β3(k)e−ikx,
0 < x ≤ a
a < x < b
b ≤ x
(44)
where k = E1/2and k′=√E − V0for E ≥ V0> 0 or k′= i√V0− E for V0> E > 0. The
α2(k)=
1
2e−ik′a
1
2eik′a
1
4e−ikb
?
sinka −
?
sinka +
k
ik′coska
k
ik′coska
?
α1(k)
α1(k)
β2(k)=
?
?
α3(k)=
(1 + k′/k)eik′(b−a)(sinka +
k
ik′coska) (45)
+ (1 − k′/k)e−ik′(b−a)(sinka −
1
4eikb
k
ik′coska)
k
ik′coska)
k
ik′coska)
?
α1(k)
β3(k)=
?
(1 − k′/k)eik′(b−a)(sinka +
+ (1 + k′/k)e−ik′(b−a)(sinka −
?
α1(k)
with α1(k) to be determined by normalization conditions (see below).
Given the full set of solutions {ψE(x)}E∈R+ for the continuous spectrum it is easy to
find the sets {ψ±
to incoming and outgoing asymptotic conditions. Using Dirac’s notation we have32
E(x)}E∈R+ of solutions of the Lippmann-Schwinger equation corresponding
?x|E+?≡
ψ+
E(x) =−1
2i
ψE(x)
β3(k)
ψE(x)
α3(k).
(46)
?x|E−?≡
ψ−
E(x) =1
2i
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1
23
45
6
t ?i
0.2
0.4
0.6
0.8
1
?AΨ?t??2
Figure 1:
|Aψµ1(t)|2, dashed line - |Aψµ′
time behaviour of |Aψµ1(t)|2.
survival amplitudes: Solid line -
3(t)|2.Insert shows short
1
23
4
t?10?5?i
-7
-6
-5
-4
-3
-2
-1
log??A?2?? 10?6
1
23
45
6
7
x
1.82125
7.02336
14.2336
10
E,?Ψ?2
Figure 2: Probability densities of approximate
resonance states ψ(n)
µj, j = 1,2,3 for a = 2, b = 3,
V0 = 10. Dashed lines - |ψ(0)
Solid lines - |ψ(9)
µj(x)|2, j = 1,2,3.
µj(x)|2, j = 1,2,3.
1234
x
0.5
1
1.5
2
2.5
?ψ?2
Figure 3: Probability densities of the approximate
resonance states for µ′
3= 17.4652 − i4.4029.
dashed line - |ψ(0)
µ′
line - |ψ(9)
µ′
Dot
3(x)|2, dashed line - |ψ(8)
µ′
3(x)|2, solid
3(x)|2.
The normalization conditions for the Lippmann-Schwinger states in Eq. (46) determines
α1(k) with the result α1(k) = (2πk)−1/2. In the energy representation the S-matrix is then
given by32
˜S(E) = −α3(k)
β3(k),k = E1/2
We now have the ingredients for the calculation of the scattering resonances and the corre-
sponding approximate resonance states. Note first that α3(k) and β3(k) can be extended to
analytic functions in the complex k plane and as a result the poles of the analytic continuation
of˜S(E) to the lower half-plane (i.e., across the square root cut along the positive real axis)
are identified with zeros of the function β3(k). For a resonance at a point z = µjin the lower
half-plane below the positive real axis we set µj= Eµj−iΓµj/2, with Eµjbeing the resonance
19
Page 20
energy and Γµjits width.
Using Eq. (27) and the expression for the outgoing Lippmann-Schwinger eigenfunctions
?x|E−?, Eq. (44), (45) and (46), the zero’th order approximate resonance states for the square
barrier problem can be calculated numerically. Considering a larger number of resonance poles
we are able to calculate higher order approximate resonance states using Eq. (30). As an
example we consider the three lowest energy resonance poles for barrier parameters a = 2, b =
3, V0= 10. These poles are located at µ1= 1.8213−i0.0023, µ2= 7.0237−i0.0564 and µ3=
14.2336−i0.8923. The zero’th order probability densities |ψ(0)
are shown as dashed lines in Fig. 2 while the solid lines on the same figure correspond to the
9’th order probability densities |ψ(9)
µj, j = 1,...,10 are taken into account in Eq. (30). We observe the significant change in the
probability density profile between the zero’th and 9’th order approximate resonance states
for the resonance µ3, whose energy is higher then the barrier’s energy, while the lower two
states are essentially unchanged. Numerical calculations show that approximate resonance
states converge in L2norm to a limiting state as a function of the order of approximation.
An example is provided in Fig. 3 which shows the probability density |ψ(n)
resonance µ′
3= 17.4652 − i4.4029, at barrier parameters a = 2, b = 2.1, V0= 10, and for the
orders n = 0,8,9. At present a rigorous criterion for the rate of convergence of approximate
resonance states as a function of order is not yet established.
Turning to a consideration of the time evolution of survival probabilities for resonances of
the square barrier model, we first recall the fact that the time evolution of the survival prob-
abilities of higher order approximate resonance states corresponding to the same resonance
pole is independent of the order and is, in fact, identical to that of the zero’th order state.
Bearing this in mind we may omit in our notation any indication of the region Σ or the order
n and set Aψµj(t) ≡ AψΣ
|Aψµ1(t)|2for the states coresponding to the lower resonance in Fig. 2 is shown as a solid line
in Fig. 1. The time evolution of |Aψµ1(t)|2follows closely an exact exponential decay law with
a decay constant Γµ1= 2|ℑ(µ1)|. This behaviour is reflected in the bound |Rµ1(t)| ≤ 0.028
on the size of the background term calculated using Theorem 5. The time development of
|Aψµ1(t)|2deviates from the exponential law at a very short time scale, as is clearly seen in
the insert in Fig. 1. The behaviour of the survival probability for the other resonances in
Fig. 2 (not shown in Fig. 1) is similar. The short time deviations from exponential decay are
related to the known Zeno effect.
The nearly exact exponential decay law of the survival probability |Aψµ1(t)|2is to be
contrasted with the time development of |Aψµ′
probability |Aψµ′
decay law in this case are evidently larger. This conforms with the results of Theorem 5 which
produces the larger bound |Rµ′
µj(x)|2, j = 1,2,3 for these poles
µj(x)|2, j = 1,2,3, where the ten lowest energy resonances
µ′
3(x)|2for the
µj(t) = Aψ(n)
µj(t). The time dependence of the survival probablility
3(t)|2for the states in Fig. 3. The survival
3(t)|2is described by the dashed line in Fig. 1. Deviations from an exponential
3(t)| ≤ 0.422.
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5 Summary
The semigroup decomposition formalism makes use of the fundamental mathematical theory
underlying the structure of the Lax-Phillips scattering theory, i.e., the functional model for
C·0 contractive semigroups, for the description of the time evolution of resonances. If the
S-matrix is meromorphic in a region Σ and is known to have resonance poles there at points
z = µ1,...,µn∈ Σ, the semigroup formalism allows for the association of a unique Hilbert
space state ψΣ
approximate resonance states, define the decomposition of matrix elements of the evolution
and are associated with its semigroup part. Theorem 5 provides an upper bound on the
size of the remaining background term. Depending on one’s knowledge of the location of the
resonance poles it is possible to calculate approximate resonance states of different orders.
Numerical calculations show that the sequence of approximate resonance states appear to
converge in L2norm to a limiting function as a function of the order. However, rigorous
criteria for the rate of convergence are needed. Another possible course of further investigation
involves the study of relations between known frameworks for the treatment of the problem
of resonances, such as the rigged Hilbert space method and the use of dilation analyticity and
the formalism discussed in the present paper.
µj(x) ∈ Hac(H), j = 1,...n with each resonance. The states ψΣ
µj(x), called
Acknowledgements
The work of Y. Strauss was partially supported by ISF under Grant No. 1282/05 and
Grant No. 188/02, and by the Center for Advanced Studies in Mathematics at Ben-Gurion
University and the Edmond Landau Center for research in Mathematical Analysis and related
areas, sponsored by the Minerva Foundation (Germany).
1P.D. Lax and R.S. Phillips, Scattering Theory (Academic Press, New York, 1967).
2V.M. Adamjan, Funkts. Anal. Prilozh. 10, 1 (1976).
3J. Cooper and W. Strauss, Indiana Univ. Math. J. 34, 33 (1985).
4R. Phillips, Indiana Univ. Math. J. 33, 832 (1984).
5C. Foias, J. Funct. Anal. 19, 273 (1975).
6D.Z. Arov, Soviet Math. Dokl. 15. 848 (1974).
7J. Sj¨ ostrand and M. Sworski, J. Funct. Anal. 123, 336 (1994).
8S. Kuzhel, Ukrainian Math. J. 55, no. 5, 621 (2003).
9S. Kuzhel, Methods Funct, Anal. Topology 7, 13 (2001).
10S. Kuzhel and U. Moskalyova, J. Math. Kyoto Univ. 45, no. 2, 265 (2005).
11C. Flesia and C. Piron, Helv. Phys. Acta 57, 697 (1984).
12L.P. Horwitz and C. Piron, Helv. Phys. Acta 66, 693 (1993).
21
Page 22
13E. Eisenberg and L.P. Horwitz, “Time, irreversibility, and unstabel systems in quantum mechanics”,
in Advnces in Chemical Physics, edited by I. Prigogine and S. Rice (Wiley, New York, 1997),
Vol. XCIX.
14Y. Strauss, L.P. Horwitz, and E. Eisenberg, J. Math. Phys. 41, 8050 (2000).
15Y. Strauss and L.P. Horwitz, Found. Phys. 30, 653 (2000).
16Y. Strauss and L.P. Horwitz, J. Math. Phys., 43, 2394 (2002).
17T. Ben Ari and L.P. Horwitz, Phys. Lett. A 332, 168 (2004).
18M. Reed and B. Simon, methods of modern mathematical physics, Vol. 3, Scattering theory (Aca-
demic Press, New York, 1979).
19Y. Strauss, J. Math. Phys. 46, 32104 (2004).
20Y. Strauss, J. Math. Phys. 46, 102109 (2005).
21B. Sz.-Nagy and C. Foias, Harmonic Analysis of Operators on Hilbert Space (North-Holland, Am-
sterdam, London, 1970).
22H. Baumg¨ artel, Rep. Math. Phys. 52, 295 (2003) and errata, Rep. Math. Phys. 53, 329 (2004).
23H. Baumg¨ artel, Rev. Math. Phys. 18, 61 (2006).
24G. Gamow, Z. Phys. 51, 204 (1928).
25C.S. Kubrusly, An introduction to Models and Decompositions in Operator Theory (Birkhauser,
Boston, 1997).
26M. Rosenblum and J. Rovnyak, Hardy classes and Operator Theory (Oxford University Press, New
York, 1985).
27N.K. Nikolski˘i, Treatise on the Shift Operator (Springer-Verlag, New York, 1986).
28Y. Strauss, Int. J. Theo. Phys. 42, 2285 (2003).
29K. Hoffman, Banach Spaces of Analytic Functions (Prentice Hall, Englewood Cliffs, NJ, 1962).
30P.L. Duren, Theory of HpSpaces (Academic, New York, London, 1970).
31C. Van Winter, Trans. Am. Mat. Soc. 162, 103 (1971).
32R. de la Madrid and M. Gadella, Amer. J. Phys. 70, 626 (2002).
22