Long range scattering with Stark effect and almost periodic potentials

(1)

J. & Math. Sci. VOL. 15 NO. (1992) 175-182

175

LONG

RANGE SCATTERING WITH STARK

EFFECT

AND ALMOST PERIODIC POTENTIALS

DENIS A.W. WHITE Departmentof Mathematics,

University ofToledo, Toledo,OH43606

Received December 3, 1990 and revisedAugust20, 1991

Abstract

Spectral and scattering theory is discussed for the Stark effect Hamiltonians

Ho

-(1/2)A

zlandH

H0

/VwhereVis alongrange perturbation. Mostsignificantly, in onedimension,and forVconsisting ofaslowlydecaying term andamalmost periodic term, thetwoHilbert spacewaveoperators

(of

Isozaki and

Kitada)

areshownto existand

becompletebyEnss’stimedependent method.

KeyPhrases. Longrange scattering, Starkeffect,almost periodic,twoHilbertspace waveoperators.

1991 AMSSubject Classification. Primary35P25;Secondary47A40,81U05.

1

Introduction

This noteconcerns quantummechanicalscatteringin the presence ofaconstant electric field. Thecorresponding Starkeffect" Hamiltonians are,

H0

-A-

zlwhere

A

(1.1)

j--I

U

Ho+

V

Ho+

V

+

V+ Vs

(1.2)

where

-z

isthepotential correspondingtothe constant electric fieldinthepositive

z

direction

(x

R

)

and

H

is aperturbationof

H0

byanotherpotential

V

+

V

+

Vs

consistingof twolongrange terms andashortrangeterm.

More

preciselyassume

Conditon

LRI.

V

C(R

)/s

realvalued and

.for

somee

>

0

IDVL()[

<

C,()

-Il/z-’

for

_{every multi-indez}

where

(z,)’

(1

+

).

Conditon LR2.

VL

0

if

the spacedimension is n

>

bet

i

n then

Vz,

C(R)

is

realvalued andboundedalong withallits derivativesand

V

a x

+

(2)

ezists asan improper Riemannintegral

for

everyz.

Condition SR.

Vs

is asymmetric operator which is

Ho-compact

and

f,

II’(x

>

)Vs(Uo +/)-all

dr.

<

oo

where

F(.)

ismultiplication

bt

the characteristic

function

of

the indicatedset.

It should be clarified that the term

VL2

which allows an almost periodic potential is only present in theone dimensionalcase. Much of the recent worlq has focusedon theone

dimen-sionalcase; see

Hislop-Nakamura" [1],

Jensen

[2],

Jensen-Yajima

[3]

and Ozawa

[4]

forexample. The methods hereare primarilymulti-dimensionaland therole of the assumption n will

be highlighted below. Almostperiodic potentials and theexistence and completeness of the ordinarywaveoperators

(of

equation

(1.6) below)

havebeenstudiedpreviously

[2].

Long

range scatteringin thissettingisrelativelynewbut there has beensomeinterestrecentlybecause there

is adiscrepancy betweenquantum and classical mechanics; theordinary waveoperatorsexist in the classicalsetting but not the quantumonewhenn and

VL2

0aswasnoted recently byJensen-Yajima

[3]

and Jensen-Ozawa

[5].

In

Theorem 2 below an additional condition on

the potentialisintroduced which assures that the two Hilbertspace waveoperators equal the ordinarywaveoperators.

In

generalone _{expects that the two Hilbert space}waveoperators are

equal,uptoaphase function,tothe Dollard’s modifiedwaveoperators; thejustification ofthis

expectation will be the subject ofafutureinvestigation. Thatresult and Theorem belowwill

preciselydelineatefor which potentials of those considered

here,

the usualwaveoperatorsexist

and arecomplete. Forthe potentials

V1

thecondition is e

>

1/2 (see

Jensen-Yajima

[3]

and Ozawa

[4]

when

n=l)

but for

Vt;

the conditionisnot obvious; seeTheorem2.

Themainresult istheexistenceand completenessofthe two Hilbert spacewaveoperators,

W

+

W+(H,

H0;d

+)

definedby

W

+/-s-lim

eitHJ+e-itH

(1.3)

where J+/- are two bounded operators on

L(R’),

to be specified and "s-lim" is thelimit in

thestrong operator topology. Notethe reversal of signinequation

(1.3)

whichisfor historical

reasons

(see

Reed and Simon’s third volume

[6,

p.

17]).

The twooperators

J+

couldequallywell be replacedby asingleoperatoraswaspreviously notedby theauthor

[7][equation

(2.10)]

and

asiscustomaryinmost discussions oftwoHilbertspace scattering.

(The

operators

J+

arenot

unique.) Isozakiand Kitada

[8]

werethefirstto introduce

J+

("time

independent

modifiers")

in this context;theyaredefinedby

f

ei"+’*(’O()d,

(1.4)

J+

r(x)

where isthe Fourier transform of

b,

dx

(2r)-n/dx

and 0+arefunctions to bechosen. The

functions 0+willbespecified in

3

belowbutroughly theyarechosenso that the commutator

HJ

+

J+Ho

issmall.

(Recall

Cook’s

method.)

Thesymbolof

HJ

+

J+Ho

is,at leastinthe

casethat

Vs

0:

___00+

(,)

"

v0(,)

+

(,0-+

1/2v0(,).

v0(,

)+ v()

+

v()

(.)

(3)

LONG RANGE SCATTERING WITH STARK EFFECT 177

Theorem 1 Assume the Conditions Litl, LHP. and Sit. Then

for

.I

4-as

defined

above

(for

appropriate

04-),

the wave operators

W4-

of (1.3)

ezist, are isometries andare complete. Moreover

H

hasnosingularlycontinuousspectrum, anditseigenvalues arediscreteand

of

finite

multiplicity.

Theorem 2 Under the hypotheses

of

Theorem 1 and in the special casen 1,

VL1

0 and

lim

VL,(x

1/2’

+ It’)dr

0

it ispossible to choose

J4-

tobe the identityoperatorin Theorem

I,

thatis

W

+/-s-lim e

itue

-*u.

(1.6)

t

Examples. The conditions of Thereto are verified if: for e

>

0 and a,

1/2

and

7

>

1/2

and b,

,

barereal,

VL(X)

sin(bx

)

+

U’(x)

where

U

is realfunction which is bound along with all its derivmtiv

(but

V

m 0 if

n

>

1.)

The conditions ofThrems and 2arelinear andmthe tof potentialsveredby theeresults form linear spe. For examplof theshort rangepotentials

Vs

sYajima

[9].

Roughly

Vs

should be

O((x)

-/-)

for

>

0 androme

>

0 and

o((x))

for

<

0.

One simple exampleindicates thatthereisinddadifferencebetwnoneandmore

dimen-sions: for

V(x,x)

sinx +sinx

andn 2 thewaveoperatom

(1.6)

do notexistbyatensor

product argument where they do if

VL(m)

sinmwhen n 1. If

V(x)

sin(x

/3)

then Threm applies but not Thereto 2.

Jenn,

[2]

using

Mourre’s

[10]

methodobtainsarult similar toThereto2 foracls of realvalu almostperiodic potentis,

r

i_

g()

e

"

()

prvia

(

+

-

(()

<

(and

ith

e

and

g

0).

Threm 2

ruir

moredifferenfibilitybut neles

enti-derivative;mrepredsely,for ehN

Threm 1,hieh indudesboth lly deeyi end

m

periodic ptenfial,ine.

2

Two

Hilbert

Space

Scattering.

The derivation ofTheorem 1.1 is broken into two main steps. The first step

(Theorem

2.1

below)

istoderivenecessary conditionson

J4-

or,in viewof

(1.4),

conditionson

04-

toconclude Theorem 1.1. The second step

(in

3)

is toconstruct

04-

satisfyingthose conditions, fromthe longrangepotential

VL.

Portionsof theproofswillbe cited from White

[7]

which will bequoted frequentlysimplyas

[W]

for brevity’s sake.

More general

potentials

Vta

areallowed in theone

dimensionalcasebecause

(4)

since it isunitarily equivalent via

exp(-iD/6)

to a HilbertSchmidtoperator

(see

Perry

[11,

Proposition

19.1]);

here

D

=-iO/cz.

Before stating theconditionson 0+/- _{it is convenient}_to_recall_{the Calder6n-Vaillancourt}

the-orem which willbe required to show that

J:

and related operators are bounded. Introduce therefore the pseudo-differentialoperator

R

for all fi

$(R"). Here

"Os-" indicates thatthe integralsareoscillatoryintegrals

(see

Kumana-go

[12]).

The theoremcanbestated in terms of certainnormsonthesymbol,p: for eachinteger k

Ipl,

,p{IOO,(z,,)l,

lD’O,(,,)l}

(2.3)

where thesup isoverall

(z,

y,

)

/R3"_and_{all or,}

,

3’ sothat 0

< I1,

I’rl

_<

2, ,nd0

_<

I1

<

2([n/2]

+

k

+

1)

and wherernistheleast integerm

>

5n/4.

The Calder6n-Vaillancourt theorem

[13]

says that thereisaconstant

C

notdependingonpsothat

IIRll

<

Clpl0

where

II"

ao

hoperatornormon

Anticipating theuseof

Enss’s

timedependentmethod

([14])

weintroducesmooth versions

of the incoming and outgoingoperators,

F(D < 0)

and

F(D

>

0)

respectively. Choose t/in

C(R)

sothat

if

>

1,

r/(,)

and

t/(,)

+

r/(-,)

1.

(2.4)

0 if

<

-1

Then thesmooth versions of the incoming and outgoing operators are

t/(-D)

and

t/(D)

re-spectively. Define further

t/,(,)

r/(,

2)

sothat if,

>

3,

(2.5)

/’(’)=

₀ _if,<l.

Thehypotheses on 0

:

can now be stated, beginning with 0- for simplicity. Thefirst

as-sumptionsaretechnical: 0-fi

G(R

x

R"),

iscomplexvalued andfor allmulti-indiciescrand thereareconstants

C

>

0,and

N

>

0sothat

IDDO-(,)I

<

c

provided

Icrl ’_>

IO0-(,)l

<

C(

/

I1)(

/

I1);

(2.e)

Ic%0-I.,

<

/2.

(2.7)

where

()

(

/

I1

)

ndm istheleast integerrnE

5n/4.

Now

acan

.-

by

(1.5).

Theshort range assumptionis

],oo

1’,(,/")’,(6/")f

l,

d,"

<

o.

(2.8)

Furthertherearetwocompactnessassumptions and here thedependenceondimension becomes apparent.

Assume,

forrt

.]i2n[I,,(=l,’),,(il,’)f

l,.

+

I,,(=l,’),,(el,’)O0-10=l,,,+,

(2.9)

(5)

RANGE SCATTERING 179

where30- denotes theimaginary partof 0-. Whenn wewrite zfor xl etc. Ifn

>

then astrongerassumptionisneeded:

,,lim[[r/(s

a)(z

b)qs()p-[,,,

+

Ir/(

a)q(zs

b)rt()V,O-[=+

+

I(

b)(

)()-I]

0

(2.0)

for all real a, b. Similarhypotheseshold for 0+ but it is more convenientto define

O+(z,)

-O-(z,-)

sothat

J+(z)=

J-()(z)

("time

reversal")

where the bar denotescomplex conjugation. Under theseassumptionswehave thefollowing theorem.

Theorem 1

Define

Ho

by

(I.I)

andlet

H

Ho+ V.

+

Vs

where

VL

V(x)

isarealvalued

function

which as an operatoracts multiplicatively, is Ho-bounded, and has

Ho-bound

strictly less than 1 and where

Vs

satisfies

Condition

SR.

If J+

and

p

areas

defined

inequations

(1.)

and

(1.5)

and

if

0

4-and

p+

satisfy the abovehypothesesthen theconclusions

of

Theorem I. are valid.

Outlineof the Proof. Theproofisby

Enss’s

timedependentmethod which hereis

adapted

toatwo Hilbert space setting appropriate forstudying longrange scattering. The methodcan

be describedasfollows. Let

H

beanyself adjointoperatoron

L2(R

")

withspectralmeasureby

E

(but H0

isdefinedby

(1.1))

and suppose

J+

bebounded operators.

Assume

further HypothesisH1. Thereisao

<

-1 sothat,

for

alla,a

>

ao

((H

+

i)-lJ

+

J+(Ho

+

i)-)y(q=Dl

-a)

arecompact.

Hypothesis H2. Foreverycompactreal interval

I,

there isan integer

N

sothat

/t

IIE(I)(HJ

+

J+Ho)h(:FD/r)y(xl/r2)(Ho

+

i)-lldr

<

oo.

Hypothesis H3. The following operatoriscompact

(H

+

i)-[J+l(-D)(J+)

+

J-l(D)(J-)’- I](H +,i)-.

HypothesisH4.

eitHo((J+)*J

:

1)e

-irate --,₀ _weaklyas _Too.

Enssmethodarguments applytoderivethepresenttheoremfrom thesehypotheses;see

[W,

Theorems 2.1 and

2.2].

The first two

hypotheses

correspond tothe standard

Enss

assumptions for theshortrangecase; H3assuresthat theoperators

J

are "almost" unitary andH4assures

that thewaveoperators W:areisometries, iftheyexist.

Checktherefore Hypotheses H1 through H4.

Suppose

n sincethe caseofn

>

was

consideredin

[W].

Onlythe outgoing

"-"

caseisconsideredbutsee

(2.11).

Let P

(H0

+

VL)J-

J-Ho

sothat

for

p-

definedby

(1.5).

Since

Vs

isH0-compact byCondition

SR,

itsuffices toshow that

(H

+

(6)

anyr

>

0, by

(2.1),

this may besimplifiedtoshowing that

(H+i)-’

Prl,(IDI/r)(Ho+i)-’tt(D-a)

canbemade arbitrarilyclose to acompact operatorby choosingrlarge enough. The proof of

this latter statement is thesame as theproofof H1 given in

[W]

but

P

thereis replaced by

P,,(IOl/"). (It

should be remarked that

[W,

Proposition3.2

(a)

and

(b)],

whichisused in the proofcitedabove, assumesthat 0 isreal valued but the proofs apply without

change

to the complex valuedcaseprovided the factor

exp(-0

+/-)

appearingintheFourierintegraloperators

istreatedaspartofthesymbol

(and

notthephase)asinthe proof of H3 below.

The proofof H2is exactlyas in

[W];

it uses thecriticalshort range assumption

(2.8)

and the Calder6n-Vaillancourttheorem.

ToverifyH3,itsuffices toshow that

(H

+

i)-’[dl(=[=D)(J+)"

t/(=l=D)l(H

+

i)-’

arecompact

(2.12)

because thesumofthese twooperatorsistheoperatorinH3. Theargumentisgiven forJ- only;

the

J+

caseis similar

(or

use

(2.11)).

First, by HypothesisH1,

H0

isroughly "J--subordinate"

to

H

ormoreprecisely

,lim

II(H

+

i)-’J-q(D)Eo(lml

>

s)l

0 by

[W,

Lemma

2.5].

Thisand

(2.)

reduces

(2.12)

toshowing

(H

+

i)-[J-tl(D/r)(J-)

q(D/r)](H

+

i)

-

arecompact

(2.13)

for all r

>

0.

(Note

H

and

H0

haveto samedomain and somay be interchanged in

(2.1).)

Theargumentof

[W]

applies

(see

also

[W,

Proposition

3.2(c)]).

Thechangeof variables in that argumentinvolvesonlythe realpart 0- of 0:

’

’(,

,

y)

+

vo(y

+

(

y),

)

d. andusesassumption

(2.9)

but theproofisessentially thesame.

The following theorem will be used for theproofofH4 and Theorem1.2both. Theorem2

Let

-

be a complex valued

function

on

R

2

satisfying

(.7)

and with

-bounded. Let j- be

defined

by

(1.4)

withphase

function

O- there replaced by

-.

If

m is the least integerrn

>_ 5n]4

and

if

i.m

I,,(,/.’),,(O,I.)(O-(,)

-(,))1-

0

then

Proof of Theorem2.2.

By

adensityargumentitsufficestoprovethestrongconvergence

onstatesof theform,

tl(Dl-a)tl(xl-b).

Recallthat outgoing states evolvingfreely constantly acceleratein thedirectionof the electric fieldexceptforanerrorterm:

I1(1

tl,(16x,/t2))e-itHrl(D,

a)rl(x,

b)ll

O(t)

-N

(see

Perry

[11,

Lemma

19.7]).

Alsomomentumistranslatedin thesame direction:

(7)

LONG RANGE SCATTERING WITH STARK EFFECT

by

[11,

Proposition

19.1]

of

Perry’s

book. Thusit isenough toshowthat lim

][(J-

-)(16x/t)(D

a

t)][

0.

andthis isfollowsfrom the assumptionson 0-

-

(see

[W,

Proposition 3.2

(a)

and

(b)]).

To completethe proof of Theorem 1, it remains tocheck Hypothesis H4.

By

Theorem 2, and assumption

(2.9)

makeitpossible toassume0-isreal valuedandin thiscasetheproofin

[W]

applies.

(There

is an errorin theproofjust before equation

(3.23) [W]:

the definitions of q andq3have been

interchanged.)

Remark.

To

extend the aboveproofton-dimensions,onewould needacriterion,replacing

(2.1)

when n

>

1, forapseudo-differentialoperatorlike

R

in

(2.2)

to becompactwithrespect to

H0

whichdoes not require thesymboltodecayto0as

3

Theorem

1.1.

TheproofofTheorem1.1 involvessimplychoosing0- andcheckingthehypothesesof Theorem

2.1. The short range assumption

(2.8)

isthecrucial to the choice.

Outline of the Proof of Theorem 1.1. Againitsufficesto consider the n andthe outgoing

"-"

case.

Moreover

theconstructionof0-

(and

thereby

J-)

was carried out in

[W]

in thecasewhen

VL2

0 andn

_>

1. Denotethat functionnowas

0i’.

₀

asthe appropriatechoiceof 0- forgeneral

VL2

and

V

0

(and

n

1);

then the appropriate

choiceof 0- in thegeneralcasewillturnout tobesimply0-

0i"

+

0.

Define

where p isa small parameter to be fixed after checking property

(2.7).

The Condition LR2

assures that the integralexists as animproper Riemannintegral and defines

₀

to be a con-tinuous function.

In

fact

₀

is infinitely differentiable and thederivatives may becomputed by differentiating under theintegral sign because ifthederivativesare computedformally by this formulathen theresultant integrals converge locally uniformly. This convergencecan be checkedbyintegratingby parts,

V

being integrated. The choice of

₀

ismadewiththe short range assumption

(2.8)

in mind.

Compute

p-(,O

[/-(OA(-)

-()()

d]

+ f(-()

-())(

1/2

+

Itisaroutinematter ofcheckingthat

₀

satisfiesthehypotheses of Theorem2.2 anditsuffices toremark thatinchecking

(2.7)

onemustchoose psufficiently

small;

pthenremainsfixed.

To

dothecheck of thesamehypothesesfor 0-

0[

+

₀

thereisonlyonenonlinear conditionthat requireschecking andthatisthe short rangecondition

(2.8).

Since

(8)

(see

[W,

equation

4.10])

this iseasily checked. O ProofofTheorem 1.2. ThisTheorem follows directly from Theorem 2.3 ifone chooses

0"-

0there,

o

Acknowledgement:

ThisresearchwassupportedinpartbyaFaculty ResearchFellowship awardedbythe University of Toledo.

References

[1]

Hislop, P.D.and Nakamura, S.StarkHamiltonians with unbounded random potentials, preprint

(1990).

[2]

Jensen,

A.Scatteringtheory

for

HamiltonianswithStarkeffect, Ann. Inst. Henri PoincrePhys. Theor.4e

(1987),

383-395.

[3]

Jensen,A.nd Yjim,K. Onthelongrangescattering

.for

StarkHamiltonians preprint (1990), IES,Aalborg.

[4]

Ozawa,T.Non-ezistence

of

waveoperators

for

Stark

effect

Uamiltonians,preprint

(1989).

[5]

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.for

Stark Hamiltonianswith slowly

decaying potentials, preprint,IES,Aalborg.

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Modern Mathematical Physics. IIIScattering Theory, New York,AcademicPress1979.

[7]

[W]

White, D. The Stark

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Modified

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[9]

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theorll

Commun.Math. Phys.68(1979),91-94.

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[13]

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1185-1187.

[14]

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.for

quantum mechanicalpotentialscattering, I. Short range potentials, Commun.Math.Phys.61

(1978),

285-291.

[15]

Simon, B.Phasespace analysts

of

simple scattering systems: eztensions

of

some work

of

Enss,