Time delay for an abstract quantum scattering process (Spectral and Scattering Theory and Related Topics)

Time delay for

an

abstract quantum

scattering

process

S. Richard*

Graduate School of Pure and AppliedSciences,University ofTsukuba,

1-1-1 Tennodai,Tsukuba, Ibaraki 305-8571, _Japan, E-mail: richard@math.univ-lyonl.fr

Dedicatedto

_Prof.

HiroshiIsozaki

on

theoccasion

_of

this sixtieth anniversary

Abstract

In this short review paper,wediscuss the concept oftime delay foran abstractquantum

scattering system. Its definition interms ofsojoumtimes is explained aswell as its identity

with the so-called Eisenbud-Wigner time delay. Necessary and natural conditionsfor sucha

constmction areintroduced and thoroughly discussed. Assumptions and statements are

pre-cisely formulated but proofsarecontained in two companionpaperswritten in collaboration

withR. TiedradeAldecoa.

1 Introduction

Heuristically, the notion of time delay in scattering theory is quite easy to understand. Given

a

reference scatteringprocess,this conceptshould indicatea

measure

of the advanceorof the delay

that asystemacquires during a slightly different scattering process.In otherwords,thetime delay

should be a

measure

ofan

excess

or adefect of timethatacertain evolutionprocessgets compared

to

an

apriori process. The paradigm example ofsuch two related systems consists in

a

classical

particle evolving either freely in

an

Euclidean space

or

inthe

same

Euclidean spacebutunder the

influence of

a

compactly supported potential.

Once this general notion isaccepted, onemight wonder how itcaneffectively be measured ?

For the paradigm example, the traditional setup consists in a series of manipulations: One first

considers a family ofboxes $B(r)$ centered at the origin and of edges equal _to $r>0$

.

One then

measures

the $ti$me $T_{r}^{0}$ spent by the free particle inside the box $B(r)$

as

well

as

the time $T_{r}$ spent

by the second particleinthe

same

box$B(r)$. $Si_{I1}ce$thetimedelayis

a

relativenotion,

one

defines

$\tau_{r}$

as

the difference between$T_{r}$ and $T_{r}^{0}$. In orderto have

a

quantity independent ofthe sizeofthe

boxes onefinally considersthelimit$\lim_{rarrow\infty}\tau_{r}$, and saysthatthisquantity,ifitexists, isthe time

delay between the twoscattering processes.

The above setupis obviously sensible and defines a rather comprehensiblenotion. However,

even

if these manipulations

are

convincing for the paradigm model, how

can

we

generalizethis

’Onleavefrom Universil\’ede Lyon; Universit\’eLyon 1; CNRS, UMR5208,InstimtCamilleJordan,43 blvd du 11

novembre 1918,F-69622 Villeurbanne-Cedex,France.Theauthor is supported by theJapan Society for the Promotion

procedure for

a more

complicated system ? Is it

even

possible to realize such

a

measure

for

an

abstract scatteringprocessand whatisthe underlyingreference system ?In the

same

line, how

can

we

define the notion oflocalisationin

a

box whenthereis

no

clear underlying

space

and

no

notion

ofboxes ?

Now, coming back to the paradigm example and assuming that the above quantity exists,

one

might also wonder if this quantity

can

be related to another measurement ? The

answer

is

yes,

and the corresponding notion $is$ the Eisenbud-Wigner timedelay [9, 31]. In fact, the identity

between the two notions of time delay

was

proved in different settings by various authors (see

[2, 3,4, 6, 8, 10, 11, 13, 14, 17, 18, 20, 23, 24, 27, 29, 30] andreferences therein), but

a

general

andabstractstatementhas

never

been proposed.

Quite recently,R. Tiedrade Aldecoa and the authorof thepresentessayintroduced

an

abstract

version for thetwonotionsoftimedelay andshowed thatthetwoconcepts

are

equal[22].The proof

mainlyrelies on

a

generalformula relating localisationoperatorstotimeoperators [21]. Usingthis

formula, these authors proved that the existence and the identity ofthe two time delays is in fact

a

common

feature of

quanmm

scattering theory. Note that

on

the way they took into account

a

symmetrizationprocedure [3,7, 10, 16, 18, 26, 27, 28,29]whichbroadlyextendsthe applicability

ofthe theory.

The aim ofthe presentpaper is to explain howthese two notions oftime delay

can

be

con-stmcted for an abstract quantum scattering system. In panicular,

we

introduce the necessary and

naturalconditions for such

a

construction.All assumptions andstatements

are

preciselyformulated

butforthesimplicity ofpresentation

we

refertothe twocompanion

papers

[21,22] for theproofs.

Infact,this paper is

an

expansion of the presentation donebyitsauthorattheconference Spectral

and scatteringtheoryand relatedtopicsorganised in February2011 in Kyoto in honor ofProfessor

H. Isozaki’s $60^{th}$-Birthday.

2 Asymptotic evolution

Inthis section

we

introducetheasymptoticsystem and thenecessalyassumptions

on

it.

Let $\mathcal{H}$ be a Hilbert space with scalar product and

norm

denoted respectively by $\langle\cdot,$ $\cdot)_{\mathcal{H}}$ and

$\Vert\cdot\Vert_{\mathcal{H}}$

.

The evolution of a quantum scattering system is defined in terms of the unitaly

group

generated by

a

self-adjoint operator $H$ in $\mathcal{H}$. One aim of scattering theory is to understand the

limits

as

$tarrow\pm\infty$ of the evolving state $\psi(t)$ $:=e^{-itH}\psi$ for suitable $\psi\in \mathcal{H}$

.

Obviously, not

all states $\psi\in \mathcal{H}$

can

be smdied and in factonly elements in the absolutelycontinuous subspace

$\mathcal{H}_{ac}(H)$ of$\mathcal{H}$withrespectto$H$

are

concemed withusual scatteringtheory.

Forinvestigatingthe longtimeasymptoticsof$\psi(t)$

one

usually looksfor anotherHilbertspace

$\mathcal{H}_{0}$ (which can also be $\mathcal{H}$ itself) and for a second self-adjoint operator $H_{0}$ in $\mathcal{H}_{0}$ such that the

element $\psi(t)$ approaches for $tarrow\pm\infty$ and in

a

suitable

sense

the elements $e^{-itH_{0}}\psi\pm$ for

some

$\psi_{\pm}\in \mathcal{H}_{0}$

.

Since in general these states do not live in the

same

Hilbert space, the construction

requires the introduction of

an

operator$J$ : $\mathcal{H}_{0}arrow \mathcal{H}$ usually called

identification

operator. For

simplicity,

we

shall consider $J\in \mathscr{B}(\mathcal{H}_{0}, \mathcal{H})$ but let

us

mention that this boundedness condition

canbe relaxed if

necessary.

Moreprecisely, giventhe self-adjoint operator$H$in the Hilbertspace$\mathcal{H}$,

one

looksforatriple

$(\mathcal{H}_{0}, H_{0}, J)$ such that the following stronglimits exist

Assuming that the operator $H_{0}$is simpler than $H$,the study of the

wave

operators $W_{\pm}(H, H_{0}, J)$

leads then to valuable informationon the spectral decomposition of$H$

.

This setting $is$ also at the

rootfor further investigationsontheevolution

group

generated by$H$and inparticular for

our

study

ofthetimedelay.

Now, let

us

call suitable

a

triple $(\mathcal{H}_{0}, H_{0}, J)$ which leads totheexistence of non-trivial

oper-ators $W_{\pm}(H, H_{0}, J)$ (aprecise condition is stated in Assumption 3). Notethat

we

arenot

aware

ofany general criterion which wouldinsure the existence of

a

suitabletriple. Furthermore, ifany

such suitable triple exists,its uniquenesscancertainlynotbeproved.However,inthe set(possibly

empty) of suitable triples, the additional conditions

on

$H_{0}$ that

we

shall introduce in the sequel

might select

an

optimalchoice between suitable triples.

Let

us

recall from the Introduction that the time delay isdefined in terms of expectations of

evolving states on afamilyofposition-typeoperators. In an abstractsetting, theexistence of such

a

family is notguaranteed by

any

means

and thus these operators have to be introduced by hands.

So, let

us assume

that there exists a finite family of mutually commuting self-adjoint operators

$\Phi\equiv(\Phi_{1}, \ldots, \Phi_{d})$ in $\mathcal{H}_{0}$ which have to satisfy two appropriate assumptions with respect to $H_{0}$

.

The first one, and by far the most important one, $is$

a

certain type ofcommutations relation. By

looking at the examplespresented in [22, Sec. 7],

one

can

get thefeeling that this assumption is

relatedto

a

certain homogeneitypropertyof

an

underlyingconfiguration

space.

However,

one

has

clearlynotintroduced suchaconcept uptonow, and this interpretation isnotbased

on

anystrong

ground.The second assumption

concems

the regularity of$H_{0}$ with respect to the operators $\Phi_{1}$ to

$\Phi_{d}$.While thefirst assumption is easilystated, thesecond

one

necessitates

some

preparations.

For any$x\in \mathbb{R}^{d}$let

us

set

$H_{0}(x):=e^{-ix\cdot\Phi}H_{0}e^{ix\cdot\Phi}$

for theself-adjointoperatorwith domain $e^{-ix\cdot\Phi}\mathcal{D}(H_{0})$.

Assumption1. Theoperators $H_{0}(x),$ $x\in \mathbb{R}^{d}$, mutuallycommute.

Clearly, this assumption is equivalent to the commutativity ofeach $H_{0}(x)$ with $H_{0}$. Now, in

order to expressthe regularityof$H_{0}wi$th respectto $\Phi_{j}$,we recall from [1,Def. 6.2.2] thata

self-adjoint operator$T$ with domain $\mathcal{D}(T)\subset \mathcal{H}_{0}$ and spectrum $\sigma(T)$ is said to be ofclass $C^{1}(\Phi)$ if

thereexists$\omega\in \mathbb{C}\backslash \sigma(T)$ suchthatthe map

$\mathbb{R}^{d}\ni x\mapsto e^{-tx\cdot\Phi}(T-\omega)^{-1}e^{ix\cdot\Phi}\in \mathscr{B}(?\{_{0})$

is strongly of class $C^{1}$ in $\mathcal{H}_{0}$. In such

a

case

and for each$j\in\{1, \ldots, d\}$, the set$\mathcal{D}(T)\cap \mathcal{D}(\Phi_{j})$

is a

core

for $T$ and the quadratic form $\mathcal{D}(T)\cap \mathcal{D}(\Phi_{j})\ni\varphi\mapsto\langle T\varphi,$$\Phi_{j}\varphi\rangle_{H_{0}}-\{\Phi_{j}\varphi,$$T\varphi\rangle_{\mathcal{H}_{0}}$ is

continuous in the topology of $\mathcal{D}(T)$. This form extends then uniquely to a continuous quadratic

form $[T, \Phi_{j}]$ on $\mathcal{D}(T)$, which can be identified with acontinuous operator from $\mathcal{D}(T)$ to its dual

$D(T)^{*}$. Finally, thefollowing equality holds:

$[\Phi_{j},$ $(T-\omega)^{-1}]=(T-\omega)^{-1}[T, \Phi_{j}](T-\omega)^{-1}$.

In the sequel,

we

shall saythat $i[T, \Phi_{j}]$ isessentially self-adjoint

on

$\mathcal{D}(T)$ if $[T, \Phi_{j}]\mathcal{D}(T)\subset \mathcal{H}_{0}$

andif$i[T, \Phi_{j}]$ isessentially self-adjoint

on

$\mathcal{D}(T)$ intheusual

sense.

Assumption 2. The operator $H_{0}$ is

of

class $C^{1}(\Phi)$, and

for

each $j\in\{1, \ldots, d\},$ $i[H_{0}, \Phi_{j}]$ is

essentially self-adjoint

on

$\mathcal{D}(H_{0})$, with its self-adjoint extension denoted by $\partial_{j}H_{0}$. The operator

$\mathcal{D}(\partial_{j}H_{0})$, with its self-adjointextension denotedby$\partial_{jk}H_{0}$. Theoperator$\partial_{jk}H_{0}$ is

of

class $C^{1}(\Phi)$,

and

_for

each $\ell\in\{1, \ldots, d\},$ $i[\partial_{jk}H_{0}, \Phi_{\ell}]$ is essentially self-adjoint on $\mathcal{D}(\partial_{jk}H_{0})$, with its

self-adjoint extensiondenotedby$\partial_{jk\ell}H_{0}$

.

Remark

2.1.

Readers

_familiar

with Mourre theory wouldhave guessedthat this assumption is

closely relatedto

a

$C^{3}(\Phi)$-type regularity condition. However, the unusual requirement

on

self-adjoinmess isdueto

our

use

_{ofafunctional}

calculus associatedwiththesesuccessivecommutators.

As shown in [21, Sec. 2], this assumption implies the invariance of $D(H_{0})$ under the

ac-tion of the unitaly

group

$\{e^{-ix,.\Phi}\}_{x\in \mathbb{R}^{d}}$. As

a consequence,

each operator $H_{0}(x)$ has the

same

domain equal to$\mathcal{D}(H_{0})$

.

Similarly, thedomains $\mathcal{D}(\partial_{j}H_{0})$ and $\mathcal{D}(\partial_{jk}H_{0})$

are

leftinvariantby the

action of the unitary

group

$\{e^{-tx\cdot\Phi}\}_{x\in N^{d}}$, and the operators $(\partial_{j}H_{0})(x)$ $:=e^{-ix\cdot\Phi}(\partial_{j}H_{0})e^{ix\cdot\Phi}$

and $(\partial_{jk}H_{0})(x)$ $:=e^{-i\tau\cdot\Phi}(\partial_{jk}H_{0})e^{ix\cdot\Phi}$

are

self-adjoint operators with domains $\mathcal{D}(\partial_{j}H_{0})$ and

$\mathcal{D}(\partial_{jk}H_{0})$ respectively. It has also been shown in [21, Lemma 2.4] that Assumptions 1 and 2

imply that the operators $H_{0}(x),$ $(\partial_{j}H_{0})(y)$ and $(\partial_{k\ell}H_{0})(z)$ mutually commute for each _$j,$$k,l\in$

$\{1, \ldots, d\}$ and each _{$x,$ $y,$}$z\in \mathbb{R}^{d}$. For simplicity,

we

set

$H_{0}’:=(\partial_{1}H_{0}, \ldots, \partial_{d}H_{0})$

and defineforeach measurablefunction $g:\mathbb{R}^{d}arrow \mathbb{C}$ the operator_{$g(H_{0}’)$} by using the d-variables

functional calculus. Similarly,

we

considerthefamily ofoperators $\{\partial_{jk}H_{0}\}$

as

the components of

a

d-dimensionalmatrixwhich

we

denote by$H_{0}’’$.

Remark2.2. Bychoosing

_for

$\Phi$ thesingle operator 1, orany operatorcommuting with _{$H_{0}$}, both

conditions above

are

_satisfied

byany suitable triple $(\mathcal{H}_{0},H_{0}, J)$

.

However,

as we

shall

see

in the

next section, these choices would leadtotrivial statememswith

no

_information

in them. In

_fact

a

criterion

_for

an optimalchoice

_for

both $(\mathcal{H}_{0}, H_{0}, J)$ and$\Phi$will beexplained in Remark3.5.

We

are

already in a suitable position for the definition ofthe sojoum time for the evolution

group

generated by$H_{0}$

.

However,

we

wouldlikefirsttolookatvarious

consequences on

$H_{0}$ of the

Assumptions 1 and2.

3

Properties

of

$H_{0}$

Inthissection

we

assume

tacitlythatAssumptions 1 and2hold andexhibit

some

consequences

on

the operator$H_{0}$. Ourfirst task istodefine values inthespectrumof$H_{0}$whichhave

a

troublesome

behaviourforscatteringtheory. Obviously,thisset

can

only bedefinedwiththe objects yetathand.

For that

purpose,

let $E^{H_{0}}(\cdot)$ denote the spectral

measure

of $H_{0}$

.

For shortness,we also

use

thenotation $E^{H_{0}}(\lambda;\delta)$for$E^{H_{0}}((\lambda-\delta, \lambda+\delta))$. We

now

introduce thesetofcriticalvalues of$H_{0}$

andstateits main properties,seealso [21,Lemma2.6]for

more

properties and details.

Definition3.1. A number$\lambda\in\sigma(H_{0})$ iscalled acriticalvalue

of

$H_{0}$

if

$\lim_{\epsilon\backslash 0}\Vert((H_{0}’)^{2}+\epsilon)^{-1}E^{H_{0}}(\lambda;\delta)\Vert_{H_{0}}=+\infty$

for

each$\delta>0$

.

We denoteby$\kappa(H_{0})$ theset

of

critical values

of

$H_{0}$.

Lemma3.2. Let$H_{0}$ satisfyAssumpfions 1 and2. Then theset $\kappa(H_{0})$ is closedand contains the

Now, the spectral properties of $H_{0}$ which

are

exhibited in the next proposition

are

conse-quences oftheexistenceofanexplicitconjugateoperator for$H_{0}$

.

Indeed,ithas been shownin [21,

Sec. 3] that for$j\in\{1, \ldots, d\}$ the expression $\Pi_{j}$ $:=\langle H_{0}\rangle^{-2}(\partial_{j}H_{0})\langle H_{0})^{-2}$ defines

a

bounded

self-adjointoperatorand that theoperators$\Pi_{j}$ and$\Pi_{k}$ commutefor arbitrary_$j,$$k$

.

Notethat

we use

thenotation $\langle x)$ $:=(1+x^{2})^{1/2}$ forany $x\in \mathbb{R}^{n}$

.

Based

on

this, it isprovedin the

same

reference

that theoperator

$A:= \frac{1}{2}(\Pi\cdot\Phi+\Phi\cdot\Pi)$

is essentially self-adjoint

on

the domain$\mathcal{D}(\Phi^{2})$

.

Then, since theformal equality

$[iH_{0}, A]=\langle H_{0}\rangle^{-2}(H_{0}’)^{2}(H_{0})^{-2}$

holds, the commutator $[iH_{0}, A]$ is non-negative and this construction

opens

the way to the smdy

ofthe operator $H_{0}$ with theso-calledMourretheory. Such an analysis hasbeen performed in [21,

Sec. 3] from which

we

recall themain spectral result:

ProposItion3.3. Let$H_{0}$ satisfy Assumptions1 and 2. Then,

$(a)$ thespectrum

of

$H_{0}$ in $\sigma(H_{0})\backslash \kappa(H_{0})$ ispurelyabsolutelycontinuous,

$(b)$ eachoperator$B\in \mathscr{B}(D(\langle\Phi\rangle^{-s}), \mathcal{H}_{0})$, with$s>1/2$, is locally$H_{0}$-smooth

on

$\mathbb{R}\backslash \kappa(H_{0})$

.

Remark3.4. It is worth noting that modulo the regularization $\langle H_{0}\rangle^{-2}$, the usual conjugate

op-eratorfor

the Laplace operator$\Delta$ in $L^{2}(\mathbb{R}^{d})$ isconstructed similarly. Indeed,

if

we choose

for

$\Phi$

the family ofpositionoperators$X=(X_{1}, \ldots, X_{d})$, then Assumptions1 and2areclearly

satisfied

andthegenemtor

_of

dilationis obtained by this procedure.

Remark 3.5. Onewould like tostressthat the

_definition

_of

theset$\kappa(H_{0})$ clearlydepends

on

the

choice

_of

thefamily

_of

opemtors $\Phi=\{\Phi_{1}, \ldots, \Phi_{d}\}$

.

Forexample$\iota f\Phi=\{1\}$, then $H_{0}’=0$ and

$\kappa(H_{0})=\sigma(H_{0})$_, and it

follows

that Proposition 3.3 does _notcontain any

information.

Thusthe

choice

_for

both asuitable triple $(\mathcal{H}_{0}, H_{0}, J)$ andthefamily

of

operators $\Phi$ should bedictatedby

thesize

of

thecorrespondingset$\kappa(H_{0})$: thesmallerthe better.

4

Sojourn

times

and

symmetrized

time

delay

In this section

we

introducethenotions ofsojoum times forthe two evolution groups and define

the symmetrizedtimedelay.Wealso state the mainresult

on

theexistence ofthe symmetrizedtime

delay under suitable assumptions

on

the scattering system. But first ofall, let

us

statethe precise

assumption

on a

triple $(\mathcal{H}_{0}, H_{0}, J)$ for being suitable. More precisely, this assumption

concems

the existence, the isometryand the completeness ofthe generalised

wave

operators.

Assumption3. Thegenemlised

wave

operators$W\pm(H, H_{0}, J)$

defined

in (2.1)exist andare

par-tialisometries with

_final

subspaces$\mathcal{H}_{ac}(H)$.

Theinitialsubspaces of the

wave

operators aredenoted by$\mathcal{H}_{0}^{\pm}\subset \mathcal{H}_{ac}(H_{0})$.In fact, it follows

from astandard argumentthat theoperator$H_{0}$ isreduced by the decompositions$\mathcal{H}_{0}^{\pm}\oplus(\mathcal{H}_{0}^{\pm})^{\perp}$ of

$\mathcal{H}_{0}$,

cf.

[5, Prop. 6.19]. Furthermore, themainconsequenceof Assumption3 isthat thescattering

operator

is

a

well-defined

unitary

operator commutingwith $H_{0}$. Notethatif$S$is

considered

from $\mathcal{H}_{0}$

into

itself, _thenthis operatorisonly

a

partial isometry, withinitial subset$\mathcal{H}_{\overline{0}}$ andfinal subset$\mathcal{H}_{0}^{+}$

.

We

now

define thesojoum times for the

quanmm

scatteringsystem, startingwith the sojoum

time forthefreeevolution$e^{-itH_{0}}$

.

Forthatpurpose, let

us

first define forany_{$s\geq 0$}

$\mathcal{D}_{s}$ $:=\{\varphi\in \mathcal{D}(\langle\Phi\rangle^{s})|\varphi=\eta(H_{0})\varphi$ for

some

$\eta\in C_{c}^{\infty}(\mathbb{R}\backslash \kappa(H_{0}))\}$

.

The set $\mathcal{D}_{\epsilon}$ isincluded in the subspace$\mathcal{H}_{ac}(H_{0})$ ofabsolute continuity of$H_{0}$, due toProposition

3.3.(a),and $\mathcal{D}_{S1}\subset \mathcal{D}_{s}2$ if$s_{1}\geq s_{2}$.We alsoreferthe readerto[21,Sec. 6]for

an

account

on

density

properties ofthe sets$\mathcal{D}_{s}$

.

Then,let_$f$be

a

non-negative

even

elementoftheSchwartz

space

$(\mathbb{R}^{d})$

equal to 1

on a

neighbourhood $\Sigma$ofthe origin $0\in \mathbb{R}^{d}$

.

Here

even means

that

$f(-x)=f(x)$

for

any$x\in \mathbb{R}^{d}$

.

For$r>0$and

$\varphi\in \mathcal{D}_{0}$,

we

set

$T_{r}^{0}( \varphi):=\int_{\mathbb{R}}dt\langle e^{-itH_{0}}\varphi,$$f(\Phi/r)e^{-itH_{0}}\varphi\rangle_{\mathcal{H}_{0}}$,

wheretheintegral hasto be understood

as

an

$i$

mproper

Riemannintegral. Theoperator$f(\Phi/r)$ is

approximately theprojection ontothe subspace $E^{\Phi}(r\Sigma)\mathcal{H}_{0}$ of$\mathcal{H}_{0}$, with $r\Sigma$ $:=\{x\in \mathbb{R}^{d}|x/r\in$

$\Sigma\}$

.

Therefore, if $\Vert\varphi\Vert_{\mathcal{H}_{0}}=1$, then $T_{r}^{0}(\varphi)$

can

be approximately interpreted

as

the time spent by

theevolving state$e^{-itH_{0}}\varphi$inside$E^{\Phi}(r\Sigma)\mathcal{H}_{0}$. Furthermore,theexpression $T_{r}^{0}(\varphi)$ isfinite foreach

$\varphi\in \mathcal{D}_{0}$,since

we

know from Proposition 3.3.(b)thateach operator$B\in \mathscr{B}(D(\langle\Phi\rangle^{-s}), \mathcal{H}_{0})$, with

$s> \frac{1}{2}$, is locally$H_{0}$-smooth

on

$\mathbb{R}\backslash \kappa(H_{0})$.

When trying to define the sojoum time for the full evolution $e^{-itH}$_,

one

faces the problem

that the$l$ocalisation operator$f(\Phi/r)$ acts in $\mathcal{H}_{0}$ while the operator$e^{-itH}$ acts in $\mathcal{H}$. The obvious

modification would betoconsider the operator$Jf(\Phi/r)J^{*}\in \mathscr{R}(\mathcal{H})$,butthe resultingframework

could be not general enough. Sticking to the basic idea that the freely evolving state $e^{-itH_{0}}\varphi$

should approximate,

as

$tarrow\pm\infty$,the corresponding evolving state$e^{-itH}W_{\pm}\varphi$_,

one

looks for

an

operator$L(t)$ : $\mathcal{H}arrow \mathcal{H}_{0},$ $t\in \mathbb{R}$, such that

$tarrow\pm\infty hm\Vert L(t)e^{-itH}TT^{r_{\pm}}\varphi-e^{-itH_{0}}\varphi\Vert_{\mathcal{H}_{0}}=0$. (4.1)

Since

we

considervectors $\varphi\in \mathcal{D}_{0}$, the operator$L(t)$ canbe unbounded

as

long

as

$L(t)E^{H}(I)$ is

bounded foranybounded subset$I\subset \mathbb{R}$

.

With such

a

familyofoperators$L(t)$, itisnamral todefine

a

first contribution forthesojoum time ofthefull evolution$e^{-itH}$ by theexpression

$T_{r,1}(\varphi)$ $:= \int_{\mathbb{R}}dt\langle e^{-itH}W_{-}\varphi,$$L(t)^{*}f(\Phi/r)L(t)e^{-itH}W_{-}\varphi\rangle_{\mathcal{H}}$.

However,anothercontributionnamrally

appears

in thiscontext.Indeed,forfixed$t$,thelocalisation

operator $L(t)^{*}f(\Phi/r)L(t)$ stronglyconverges to $L(t)^{*}L(t)$

as

$rarrow\infty$, butthis operatormight be

rather different from the operator 1. As

a consequence, a

part of theHilbert space might be not

considered$wi$th thedefinitionof$T_{r,1}(\varphi)$. Thus, a secondcontribution forthesojoum time is

$T_{2}(\varphi)$ $:= \int_{\mathbb{R}}dt\langle e^{-ttH}W_{-}\varphi,$$(1-L(t)^{*}L(t))e^{-itH}W_{-}\varphi\rangle_{\mathcal{H}}$

.

The finitenessof$T_{r,1}(\varphi)$ and$T_{2}(\varphi)$ is proved under

an

additionalassumption in Theorem 4.1

below. The term $T_{r,1}(\varphi)$ can be approximatively interpreted as the time spent by the scattering

scattering state $e^{-itH}W_{-}\varphi$inside the time-dependent subset $($1 $-L(t)^{*}L(t))\mathcal{H}$ of$\mathcal{H}$

.

If$L(t)$ is

considered

as

atime-dependent quasi-inversefor the identificationoperator$J$(see [32, Sec. 2.3.2]

for the related time-independent notion ofquasi-inverse), then the subset $(1 -L(t)^{*}L(t))\mathcal{H}$

can

be

seen

as an

approximate complement of$J\mathcal{H}_{0}$ in $\mathcal{H}$ attime $t$. When $\mathcal{H}_{0}=\mathcal{H}$,

one

usually sets

$L(t)=J^{*}=1$, andthe term $T_{2}(\varphi)$ vanishes. Within this general framework, the total sojoum

timefor thefull evolution$e^{-itH}$ is given_by

$T_{r}(\varphi):=T_{r,1}(\varphi)+T_{2}(\varphi)$ .

Since both sojoum times have

now

been defined, the definition of the time delay should be

athand. However, let

us

firstconsiderthefollowing dilemma. For

a

given state $L(t)e^{-itH}\psi$ with

$\psi\in \mathcal{H}_{ac}(H)$, which

one

is the correct free evolution state: is it $e^{-itH_{0}}\varphi_{-}$ with _{$W_{-}\varphi-=\psi$}

which is agood approximation for$tarrow-\infty$_, oris it $e^{-itH_{0}}\varphi+$ with _{$W_{+}\varphi+=\psi$} which is also a

good approximation but for$tarrow+\infty$ ? Obviously, both states have to be takeninto account, and

therefore

we

say

that

$\tau_{r}(\varphi):=T_{r}(\varphi)-\frac{1}{2}\{T_{r}^{0}(\varphi)+T_{r}^{0}(S\varphi)\}$,

is the symmetrized $ti$

me

delayofthe scattering systemwith incoming state

$\varphi$. This symmetrized

versionof the usual time delay

$\tau_{r}^{in}(\varphi):=T_{r}(\varphi)-T_{r}^{0}(\varphi)$

is knownto be the onlytimedelay havingawell-defined limit

as

$rarrow\infty$for complicatedscattering

systems (seeforexample [3, 7, 10, 12, 16, 18, 22,25, 26, 27]).

Thelast assumption isaconditiononthe speedofconvergence of thestate$L(t)e^{-itH}W\pm\varphi\pm$

to the corresponding states $e^{-itH_{0}}\varphi\pm$

as

$tarrow\pm\infty$. Up to now, only the

convergence

to $0$ ofthe

$no$of the differenceof these stateshad beenused,

_cf.

(4.1).

AssumptIon4. For each$\varphi\pm\in \mathcal{H}_{0}^{\pm}\cap \mathcal{D}_{0}$

one

has

$\Vert(L(t)W_{-}-1)e^{-itH_{0}}\varphi-\Vert_{\mathcal{H}_{0}}\in L^{1}(\mathbb{R}_{-}, dt)$ and

$\Vert(L(t)W_{+}-1)e^{-itH_{0}}\varphi+\Vert_{\mathcal{H}_{0}}\in L^{1}(\mathbb{R}_{+},dt)(4.2)$

NextTheorem shows theexistenceof the symmetrizedtimedelay.Theapparently large

num-ber of assumptions reflects nothing

more

but the need of describing the

very

general scattering

system;

one

needs hypotheses

on

the relation between $H_{0}$ and $\Phi$, a compatibilityassumption

be-tween$H_{0}$ and$H$, conditions

on

thelocalisation function$f$andconditions

on

the state $\varphi$

on

which

thecalculations

are

performed.

Theorem 4.1. Let$H,$ $H_{0}J$ and $\Phi$ satisfy Assumptions 1 to 4, and let

$f$ be a non-negative

even

element$of\ovalbox{\tt\small REJECT}(\mathbb{R}^{d})$equalto1 on aneighbourhood

of

the origin$0\in \mathbb{R}^{d}$

.

Then,

for

each$\varphi\in \mathcal{H}_{0}^{-}\cap \mathcal{D}_{2}$

satisfying $S\varphi\in \mathcal{D}_{2}$, the sojoum time $T_{r}(\varphi)$ is

finite for

each $r>0$ and the limit$\lim_{rarrow\infty}\tau_{r}(\varphi)$

exists.

Remark 4.2. All the assumptions in the above statemem are mther explicit except the one on

$S\varphi\in \mathcal{D}_{2}$

.

Indeed, such

a

propertyis relatedto the mapping properties

of

thescattering operator

andthis assumption isnotdirectly connectedto the other conditions. Letussimply mention that

one

usuallyprovessuchapropertybystudyinghigherorderresolventestimates.

In the next section,

_we

show that the time delay $\lim_{rarrow\infty}\tau_{r}(\varphi)$

can

be related to another

5

Time

operator and

$Eisenbud\cdot Wigner$

time

delay

We

now

define

a

timeoperatorfor theoperator$H_{0}$ andrecall

some

ofits properties from [21]. For

that

purpose,

one

needstoconstmcta

new

function$R_{f}$fromthe localisation function$f$introduced

above. Thisfunction

was

already smdiedandused,in

one

form

or

another,in [10, 21, 22, 28,29].

Thus, let

us

define $R_{f}\in C^{\infty}(\mathbb{R}^{d}\backslash \{0\})$by

$R_{f}(x):= \int_{0}^{\infty}\frac{d\mu}{\mu}(f(\mu x)-\chi_{[0,1]}(\mu))$

.

The followingproperties of$R_{f}$

are

provedin[29,Sec.2]:$R_{f}’(x)= \int_{0}^{\infty}d\mu f’(\mu x),$$x\cdot R_{f}’(x)=-1$

and $t^{|\alpha|}(\partial^{\alpha}R_{f})(tx)=(\partial^{\alpha}R_{f})(x)$, where $\alpha\in N^{d}$is

a

multi-index and$t>0$

.

Furthermore, if$f$is

radial,then $R_{\int}’(x)=-x^{-2}x$

.

Now, the nextstatementfollows from [21,Prop.5.2] and [21, Rem. 5.4].

Proposition

5.1.

Let$H_{0}$ and$\Phi sati\phi$Assumptions1and2, and let$f$bethe

localisationfunction

introducedabove. Thenthemap

$t_{f}:\mathcal{D}_{1}arrow \mathbb{C}$, $\varphi\mapsto t_{f}(\varphi):=-\frac{1}{2}\sum_{j=1}^{d}\{\langle\Phi_{j}\varphi, (\partial_{j}R_{f})(H_{0}’)\varphi\rangle_{\mathcal{H}0}+\langle(\partial_{j}R_{f})(H_{0}’)\varphi, \Phi_{j}\varphi\rangle_{\mathcal{H}_{0}}\}$,

is

_{well-defined.}

Moreover, thelinearoperator$T_{f}$ : $\mathcal{D}_{1}arrow H_{0}$

defined

by

$T_{f} \varphi:=-\frac{1}{2}(\Phi\cdot R_{f}’(H_{0}’)+R_{f}’(_{\overline{|}H}H\eta_{0}’)\cdot\Phi|H_{0}’|^{\sim 1}+iR_{f}’(\mu_{1}^{H’}0)\cdot(H_{0}^{\prime\prime T}H_{0}’)|H_{0}’|^{-3})\varphi$ (5.1)

satisfies

$t_{f}(\varphi)=\langle\varphi,$$T_{f}\varphi\rangle$

for

each $\varphi\in \mathcal{D}_{1}$

.

Inparticular, $T_{f}$ is a symmetric opemtor $\iota f\mathcal{D}_{1}$ is

densein$\mathcal{H}_{0}$

.

Clearly, Formula(5.1)_israthercomplicated and

one

couldbetemptedtoreplace itby the

sim-pler$fomiula-\frac{1}{2}(\Phi\cdot R_{f}’(H_{0}’)+R_{f}’(H_{0}’)\cdot\Phi)\varphi.$Unfomnately,

a

preci

se

meaningof this expression

isnotavailable ingeneral, andits full derivation

can

only be justified in concrete examples.

Before stating the main result of thissection, let

us

recall

some

properties ofthe operator$T_{f}$,

and referto [21, Sec. 6] for details. In the form

sense on

$\mathcal{D}_{1}$ the operators $H_{0}$ and$T_{f}$ satisfythe

canonical commutation relation

$[T_{f}, H_{0}]=i$

.

Therefore, since the group $\{e^{-itH_{0}}\}_{t\in \mathbb{R}}$ leaves $\mathcal{D}_{1}$ invaniant, the following equalities hold in the

form

sense

on$\mathcal{D}_{1}$:

$\langle\psi,$$T_{f}e^{-itH_{0}}\varphi\rangle_{\mathcal{H}0}=\langle\psi,$ $e^{-itH_{0}}(T_{f}+t)\varphi\rangle_{\mathcal{H}0}$ ,

andthe operator$T_{f}$satisfies

on

$\mathcal{D}_{1}$ theso-called infinitesimal Weyl relation intheweak

sense

[15,

Sec. 3].Note that

we

havenotsupposedthat $\mathcal{D}_{1}$ isdense.However,if$\mathcal{D}_{1}$ isdensein $\mathcal{H}_{0}$, then the

infinitesimal Weyl relation inthestrong

sense

holds:

$T_{f}e^{-itH_{0}}\varphi=e^{-itH_{0}}(T_{f}+t)\varphi$, $\varphi\in 9\text{ノ_{}1}$

.

(5.2)

This relation, _{also known}

as

$T_{f}$-weak Weyl relation [19,Def. 1.1], has deep implications

on

the

tothe Hamiltonian$H_{0}$. Moreover, being

a

weak version of the usual Weyl relation, Relation (5.2)

alsosuggests that the spectmm of$H_{0}$ may not differtoo much from

a

purely absolutely

continu-ous

spectmm. Since these propertieshave been thoroughly discussedin [21, Sec. 6],

we

referthe

interestedreaderto that reference.

Next theorem$is$themain result of[22],comments

on

it

are

providedafteritsstatement.

Theorem

5.2.

Let $H,$ $H_{0}J$ and $\Phi$ satisfy Assumptions 1 to 4, andlet

$f$ be

a

non-negative

even

element

_of

$(\mathbb{R}^{d})$equalto1 on aneighbourhood

of

theorigin$0\in \mathbb{R}^{d}$. Then,

for

each$\varphi\in \mathcal{H}_{0}^{-}\cap \mathcal{D}_{2}$

satisfying$S\varphi\in \mathcal{D}_{2}$

one

has

$\lim_{rarrow\infty}\tau_{r}(\varphi)=-\langle\varphi,$ $S^{*}[T_{f},S]\varphi\rangle_{\mathcal{H}_{0}}$, (5.3)

with$T_{f}$

defined

by(5.1).

The above statementexpresses the identity of the symmetrized time delay (defined in terms

of sojoum times) and the Eisenbud-Wigner time delay for general scattering systems. The l.h.$s$

.

of (5.3) is equal to the symmetrized time delay ofthe scattering system with incoming state $\varphi$,

in the dilated regions associated with the localisation operators $f(\Phi/r)$

.

The r.h.$s$

.

of(5.3) is the

expectation value in $\varphi$ ofthe generalised Eisenbud-Wigner time delay $operator-S^{*}[T_{f}, S]$

.

It

clearlyshows that

once

suitableand namral conditions

are

assumed, then thenotionoftimedelay

existswhateverthescattering system is.

Letus finally mentionthatwhen $T_{j}$actsinthe spectral representationof$H_{0}$asthe differential

operator$i \frac{d}{dH_{0}}$, which

occurs

in mostof thesituationsofinterest(seeforexample [21,Sec.7]), one

recoversthe usualEisenbud-WignerFormula:

$\lim_{rarrow\infty}\tau_{r}(\varphi)=-\langle\varphi,$ $iS^{*} \frac{dS}{dH_{0}}\varphi\rangle_{?t_{0}}$.

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