Finite eigenfuction approximations for continuous spectrum operators

Internat. J. Math. & Math. Sci. VOL. 16 NO. (1993) 1-22

FINITE

EIGENFUNCTION APPROXIMATIONS

FOR CONTINUOUS SPECTRUM OPERATORS

ROBERT M. KAUFFMAN

Department

ofMathematics

UniversityofAlabama at Birmingham

Birmingham Alabama35.94

(Received January 23, 1992)

ABSTRACT. Inthispaper, we introduce a new formulationofthetheory ofcontinuousspectrum eigenfunction expansions for self-adjoint operators and analyze the question of when operators may be approximated in an operator norm by finite sums of multiples of eigenprojections of multiplicity one. The theory is designed for application to ordinary and partial differential

equations; relationships between the abstracttheory and differential equations areworked outin

the paper. One motivation for the study is the question of whether these expansions are

susceptible to computation on a computer, as is known to be the casefor many examplesinthe

discrete spectrum case. The point of the paper is that continuous and discrete spectrum eigenfunction expansions aretreated bythe sameformalism; bothare limits inan operatornorm

of finite sums.

KEY

WORDS

AND PHRASES.

Continuous spectrum eigenfunction expansion, self-adjoint operator,ordinary differentialoperator,partial differentialoperator, spectraltheorem.

1991

AMS SUBJECT

CLASSIFICATION

CODES. Primary

47A70,

47E05, 34L10, 35P10.

0. INTRODUCTION

Eigenfunctionexpansionsmaybe consideredas anabstractionof theideaof approximating

complicated waves by finitely many standing waves. For discrete spectrum

eigenfunction

expansions associated with a self-adjoint operator

H

in a Hilbert space

L2(X,p)

the rate of convergence of the expansion has been the subject ofagreat dealof research.

In

thecontinuous

spectrum

case,

sums are replaced by integrals, and the question ofwhether theintegral can be approximated by a finite sum has not been studied. This paper begins such a study; first, however, we indicate whythe questionisimportant, and what sort ofanswers welook for. Itis

helpfultotakeanaivelook atthe method ofseparationof variables,or

eigenfunction

expansions.

Onepurposeofaneigenfunctionexpansionis toconvertcontinuous

data,

such asfunctions,

into elements of

:n,

vectors formed from the coefficients ofthe func.tion in the expansion.

A

problem, such as a partial differential equation, in a function space is then transferred to

:n,

solved there

(partial

differential equations often transform into ordinary differential equations

2 R.M. KAUFFMAN

transformingback, that is,bysumming theeigenfunctionswiththerecalculated coefficients. This involvesacertainerror. The errorismeasured using two Hilbertspaces andZ.

An

eigenprojection in

B(,Z),

the bounded operators from into

Z,

of a self-adjoint

operator

H

in a Hilbert space

L.

is an operator in

B(Y,Z)

of the form

P()= F()F,

where

FEZ

c ’,

and whereHtakesadenselocallyconvexvectorspace W containedin continuously

intoitself, and

H’F

,F. Ideally, theoperatorwhichsolves theoriginal problemis thelimit in

B(Y,Z)

of finite sums of multiples of eigenprojections. We call this property the discrete

approximationproperty,for the remainderofthis introduction.

If the discrete approximation property

holds,

the eigenfunctions used to perform the expansion do not have to berecalculated foreach new functionbeing expanded, and only finitely

many coefficients must be calculated.

In

other words, up to a certain error, functions become elementsof C

n,

and semigroups generated bytheoriginal self-adjointoperator becomesemigroups

ofdiagonal matrices. Continuousspectrumexpansions haveat presentnosuchtheory; these are modelled on theinverseFourier

transform,

so insteadoffinitesumsone must workwithintegrals

which in

general

only converge inthe mean, and

convergence

in operator normis not discussed.

One consequence of this is that the theory of continuous spectrum eigenfunction expansions

appearsto have no computationalsignificance; it seeminglycannot be put onto a computer. This

situation, which if true would lead to problems of whether the theory is well-posed in any reasonable

sense,

contradicts the intuition gained from the Fourier

transform;

the Fourier

transformis well-knowntobecomputationallysignificant. Thepurpose ofthispaperis to begin

a continuous spectrum theory modelledon the discrete spectrum

case,

where finite sums appear

instead ofintegrals.

In

ordertodo this,it has been useful toreformulate the existingtheory of

these expansions. Moreabout this reformulation will be given laterinthis introduction.

The discrete approximation property is shown in section 3 to follow if the operator in question is a convergent operator-valued integral, in

B(Y,Z),

of eigenprojections, which in the

continuous spectrum case must go from aspaceY smaller than

L.

to a space

Z

large enough to

containtheeigenfunctions.

Hence

we muststudywhen the expansionissuch anintegral. This is

shown in section 3 to be true when the measure of A is finite, with respect to an invariant measure dependingfor acyclic subspaceonlyuponZ. This measure is the onewhich normalizes theeigenfunctionsinZ.

ForSturm-Liouvilletheory forthe Dirichlet problem on a

fini’te

interval, with

H

and

Z

L

(R),

the sort of

convergence

westudyis well known tooccur, asitdoesinmanyother discrete

spectrum problems. Even indiscretespectrumproblems, however, the calculationofappropriate

spaces and

Z

isoftennontrivial.

In

thispaper,inthediscrete or continuousspectrum

case,

they

are calculated using a priori estimatesonthe domain of H.

Eigenfunctions

satisfythe equation

H’F

,F inacertaindualspaceandaremembersof Z.

To

studythe discrete approximationpropertyfor an arbitrary boundedcontinuousfunction of

H,

we show thatit is sufficient to studytheproperty for thespectral projections

P(A)

for

H,

correspondingtoBorel sets

A. We

showinsection 5thatthe discrete approximation property for

p(A)

is equivalent tothequestion of whether

p(A)

is compactin

B(Y,L.),

a questionofinterest in itsownright.

We

obtainresults whichgive compactness forconcreteexamplesinspaces where

it isotherwisenotknown. Thisshowsthatthetheoryhasconsequenceswhichreachbeyond itself,

EIGENFUNCTION APPROXIMATIONS FOR CONTINUOUS SPECTRUM OPERATORS 3

the separation of variables scheme discussed

earner

works.

It is not quite accurate tosay that existing theoriescompletely ignore these questions.

A

little bit of thought will convinceone that for a bounded set

A,

the inverse Fourier transform

representationofthespectralprojection

P(A)

fortheself-adjoint operator associatedwith

id/dx

has discreteapproximationpropertiesfrom, say, Y

{0:

0/w

E

L2}

toZ

{F:

wF

E

L2}

where w

is a well-behavedL

2 function. From the equivalence, shown in section 5, of the approximation

property for

P(A)

with the compactness of

P(A)

in

B(Y,Z),

together with known results about SchrCdinger operators, it is not difficult to establish the same properties for the Schrdinger operator with a well-behaved potential, where

A

is a bounded set of energies.

However,

in situations likethis, the compactness of thespectralprojection isknown beforehand, and maybe usedas in Section5 toproducethe approximationproperty.

A

moreinterestingproblemisthat of of unbounded sets

A.

With the

Y

and Z above, for theinverse Fourier transform, itturns out that someunboundedsets

A

have this propertyand others do not. The compactness of

P(A)

is

only known as aconsequence of thetheory. The results ofthis paper are oriented toward the

study ofwhichsets

A

have this property.

As

anexample, thegeneralSturm-Liouville case on a half-lineisstudied in section4. Much sharper results for

short-range

potentials are given by D.

B.

Hintonandtheauthorin

[4],

usingtheresultsofthispaper. Examples frompartial differential equations also fit easilyintotheformalizationofthispaper and explicitexamples are given.

Here

the results arelesssharpunless onerestrictsto asinglecyclicsubspace.

The heart of the paper is the operator valued integrals ofSection 3, together with their

relation to the approximation problem.

In

section 5, we show the equivalence of the two

problems, compactnessand approximation. Usingthe resultsofsections 3,4 and5of thepaper,

wesee that certainspectral projections

P(A)

aecompactasoperatorsbetweenspaceswherethey

arenot alreadyknown tobecompact bya prioriestimates.

A

simpleexampleisgivenin section

4,whichisaboutsecondorder ordinary differentialoperators.

Thetheory ofcontinuous spectrum eigenfunctionexpansions is averyold

one,

going back to Gelfand’s workinthe 1950’s. Thebook ofBerezanskii

[1]

is afundamental

reference,

butit is

difficult to extract specific informationfrom such ageneral theory. TheworkofSimon

[8],

which

isfunctional-analyticthoughit isspecifically slanted towardSchrSdingeroperators,isa clearand rigorous approach to the theory with a lot of specific information. The paper of

Poerschke-Stolz-Weidmann

[5]

ismoregeneral,and also has more

elenentary

proofs. Thispaper

has beenfollowedupbyPoershke and Stolz

[6],

who give applications oftheirresults to scattering

theory.

With such a

large

and excellent literature, why give yet another approach to the whole

theory? We

do

so,

partlyto obtainthecrucialassertion

iii)

of

Lemma

1.6, whichweneed for the

basicproblems discussedearlier, but also to be able to analyze the expansionin aformat based

simply upon a priori estimates onthe domain of powers ofthe self-adjoint operator

H

whichis

being decomposed, so as to makethe results as concrete as possibleforapplications to differential equations. The question of what is needed about an a priori estimate in order to do this is

answered in the paper. Our proofs are self-contained, since once the formalism is set up and

Lemma 1.6 is proved, the inverse Fourier transform

(Lemma 3.4)

and the Fourier transform

(Theorem

1.8)

followquite directly fromthespectral theorem; to attempt to invokeother results would introduce technical difficulties. Theestimatesontheeigenfunctionsinthispapercontained inAssertion

ii)

of Theorem 1.8 do not follow

(at

least

directly)

from other

results;

and as was

4 R.M. KAUF FMAN

inLemma3.4 is not studiedat allin existingliterature. On the otherhand, our hypothesesare

differentfrom those of otherapproachessuch as

[5]

and

[8];

for example,our theory alsodemands more smoothness on the coefficients when applied to differentialequations, as we discuss below. Therelationshipbetween this work andthatof

[5],

[6]

and

[8]

isaninterestingquestion forfuture

research.

The formalism of our theory of continuous spectrum expansions depends on the

introduction of a locally convex space W with certain properties, such that H takes W

continuouslyintoitself. Itis neededinordertohave a corewhere all operations makesense, and

from whichestimatesmaybeextendedbythe closed

graph

theorem.

It

alsoallows us tosaywhat aneigenfunction

;

it isjust anelement of

W’

suchthat

H’F

AF.

This, togetherwithregularity theorems for thedomain of

H,

if

H

is a differentialoperator,is what

turns

an abstractly defined eigenfunctionintoaconcrete object such as a smooth function.

For

example, ifW

C(fI),

and fis anopen subsetofaC(R)

manifold,and

H

is

generated

byahypoellipticdifferential expression, then

W’

is thespace of distributions,so that if

F

W’

and

H’F

AF,

then

F

isa C(R)

function. If

the operator

H

is,for example,associatedwitha Dirichletproblem, smootheness of

F

isneeded to show that

F

vanishesontheboundaryof fl. Itshould benoted that the smoothness ofFdoes not follow only fromthefact that

F

W’,

which isimpliedby virtually any theory,butfromthisfact

ogether

withthefactthat

H’F

AF.

Of course, there aremany approaches to these expansions whichimply, forexamplefor SchrSdingeroperators, that thegeneralizedeigenfunctions satisfythe

differentialequationina distributional sense andhenceclassically, but theseassertions areshown as consequences ofspecific properties ofthe examples being studied. Sincesuch assertions are

necessary forapplications, we buildthemintothetheory, producingamorepowerfulstructure.

Motivated bythe abovediscussion,inexamples studiedin this paper weoften takeW to be

C(fl);

thiscauses us to assumesmoothnessof thecoefficientswhen

H

is a differentialoperator.

However,

other choices of W would perhaps allow more general coefficients. This is another subject for further work.

The author has beenfortunate enough tohavemany discussions ofthistheorywithmany

different mathematicians over a periodofsomeyears.

He

would likeespeciallytothank Christer Bennewitz, RainerHempel, and

Don

Hinton,

although

discussions with a numberofothers have

alsobeenvery helpful. Hewould also like to thank

W.

D.

Evans,

University

College,

Cardiff, and the BritishSERCfor support during the author’s very pleasant

four-onth

stayinCardiffinthe spring of 1987, when this paper wasbegun,and

Peter

Hislop and the University of Kentucky for

theirhospitalityintheSpringsemesterof1991,when the researchfor thepaperwas finished.

1.

BASIC

FORMALISM

AND

L

2ESTIMATES

In

this section we develop the basic formalism andeigenvector estimates forour theory.

We

shall need to introduce some basicspaces

W

and

W’. W

is containedin the domain ofthe

self-adjoint operator

H,

and

W’

is its dual spaceunder acertain

topology. One

maythinkof

W

aslike

C(IR

n)

andW’ asthe spaceofdistributionson

in;

alsoonemaysometimes wishtothinkof

W

as therapidly decreasing functions and

W’

as the tempereddistributions.

In

order to handle

the casewhere W

C(In),

weneed to assumethat

W

is an inductive limit of Frechet spaces,

rather than a Frechet space itself, since

C(i

n)

under the usualtopologyisnot metrisable.

(See

Proposition

5,

p. 125, Robertson and Robertson

[7].)

The purpose of these topological vector

EIGENFUNCTION APPROXIMATIONS FOR CONTINUOUS SPECTRUM OPERATORS 5 anapplication of the closedgraphtheorem to obtain aprioriestimates from assertions about the

domainofH.

Notation 1.1.

Let

b

beaHilbert space, and let

H

be aself-adjointoperatorwith domain a

dense subspace of

b

and range contained in

.

Ife E

D,

let S_e denote the closed linear span of

{P(A)e

A

isaBorel subset

of},

where for any Borel set

A,

P(A)

is the spectral projection

associated with

A

bythespectral theorem. Let

e(A)

[P(A)e,e],

where

[,

denotes theinner

product of

D.

(In

thispaper, willalways be

L2(X,p),

wherepisa positivemeasureon

X).

Note

that

e

is a positive Borel measure on

,

such that

e()

Ilell

2.

Note

also that the restrictionof

H

to S_eis a self-adjoint operatorwhich is unitarily equivalent to the operator in

L2(e)

which maps

f(A)

to

Af(A).

Assumption1.2. Throughout thepaper,weshallassume thefollowing hypotheses:

i)

H

is a self-adjoint operator with domain a dense subset of

I

and range contained in

f

L2(X,p),

where X is a locally compact Hansdorff space, and is a positive regular Borel measureonXsuch that themeasureofeverycompact setisfinite;

ii)

W

is theinductivelimitofa sequence

{Vn}

of separableFrechet spaces such that for each n, V_n is algebraically and topologically contained in or equal to

Vn+l;

(hence

W is

complete, by

Prop.

3, p.128,

[5]); (note

that a subbase for the topology ofW is the set of all

absolutely convexsubsets U ofW suchthat Uf]V

n is open in Vn for every n; recall that W is metrisableifandonlyifforsome

M,

V

n V

M

forn

>

M,

by Prop. 5,p. 129,

[7]);

iii)

W’

is the dualof

W,

andW’isgiven thetopology

oW’,W)

of pointwise convergence

on

W;

(recall

that aneighborhood subbase about 0for thistopology is the set ofneighborhoods

UCx,)

{F: IFCx)l

<

});

iv)

w

c

domainHandHisa continuouslineartransformation fromWinto

W;

v)

W iscontained and densein

LI(X,)IL2(X,

and the identity mappingfrom W into

L

l(X,p)

and

L2(X,fl

iscontinuous;

vi)

for any openset in

X,

ifC

c(X

denotesthe continuous complex-valuedfunctions of compact support in

X,

and if is any element in

Cc(X

which is’supported in

F,

there is a

sequence

{n

}

ofelementsof

W,

each supportedin

r,

such that

Cn

convergesin

L2(X,p

to

.

Remarks:

We

shall sometimesassume thefollowing estimate; we shall explicitly statethis

assumption eachtime.

Estimate1.3.

(an

a priori

estimate)

Thereexists a 1-1 continuouslineartransformation

B

from W onto W and a positive function fE

L2(X,p),

such that multiplication by f maps W

continuouslyinto

W

andsuch that the linear transformation

B’

hasthe property that thereexists apositiveinteger N such that

{B’}/f

L(R)(X,p)

for all inthe domain of

H

N.

(Note

thatby

v)

ofAssumption1.2,

L2(X,p

isnaturallyembeddedin

W’).

Reraar.

Wenowmaketheinitialdefinition of theeigenfunctions, aslinear functionalson a dense subspace of Wover the rationals. The work consists of showing that they belongto a natural space.

Note

that W has a countable dense set by hypothesis

ii)

above. Let S’ be a

6 R.ll. KAUFFIIAN

Notation,: Let e E h. Let

Ce

be the unitary mapping from S

e onto

L2(ae)

such that

(e(e)=

1, and such that

Ce(H)(,)= ,e()()),

for all in

D(H)0Se,

and such that

(e(P(A))

x(A)(e(),

where

x(A)

denotes thecharacteristic functionof the Borel set A. Let U

e

CeoP(Se

),

where

P(Se)

denotes the orthogonal projection onto S

e.

Note that the existence

anduniqueness of

e

follows from the spectral theorem. Let S S’

+

HS’. Foreach

i

E

S,

select

aneverywheredefined representativeof

Ue(Bi

).

Denotethe above representative_{by gi"}

Definition1.4. For each

,

e

,

define

Z,,e

on S by

Z,,e(i)

gi(,),

so that ifU_eis the mappingdiscussed abovewhich arisesfrom thespectral theorem, then

Z,,e(i)=

Ue(Bi)(

for almost every

,

withrespect to a

e.

Note that, ifA

0 is the complement of the set of

,

suchthat

,,e

isalinearfunctionalonSover therationals, then

ae(0)

0.

Lemma

1.5.

If

thereexistsapositive constant

M

such that

II(n’0)/fll(R)

<_

U(ll011

+

HNolI)

1/2 for

aU

0 the domain

of

H

N,

wherefisas in Estimate 1.3,

then

for

almost everyx with respect top thefollowing is true:

for

any orthogonal set

{ei}

inthe

d’omain

of

H

N,

such thatHe

i/s

alsoanorthogonal

set,

and such that

Ileill

2

+

IlaNeill

2

x for

aU

i,

then(ii=l

IB’eil2(x))

1/2

<_

Mr(x).

Proof:

It can beproved bya technique like that ofWeidmann

[9],

p. 140, that ifT is a

bounded operator from a separable Hilbert space h into L

(X,#)

and

IITII

is the associated

operator

norm,

thenfor almost everyx with respect to #, it istruethat for anyorthonormal set

{ei}

inh,

(ZilTei(x)12)

1/2

<_

IITII.

The ideaoftheproofisto construct a mapping

Q

from

X

into

h such that

Tg(x)

[Q(x),g].

Then one canseefrom the hypothesis thatfor almostevery x,

Q(x)

has norm less than or equal to

IITII.

Thenfor each fixed x, the Schwartz inequality gives the

desired result. To make thisproofrigorous demands careful attentionto sets of measure 0 with

respect to

.

Now if we let

Tg

be

(B’g)/f,

and h be the domain of

H

N with norm

Ilgll- Ilgll

/

IIHNglI,

the resultfollows.

Anotherwayto constructthe mapping

Q

whichdoes not pay such carefulattentiontosets of measure0 isdue toC.Bennewitz andthe author:

Note that by the Gdfand representation theorem for commutative Banach algebras L

(X,p)

is isomorphic and isometric as a Banach algebra to the algebra

C(Y)

where Y is the maximal ideal space of the Banach

algebra

L(R)(X,p),

a compact Hausdorff space.

Let E

be the

isometry from

L(R)(X,p)

onto

C(Y).

Define to be theoperatorET. Then ateveryy

e

Y,

define

(y(g)

tobe

ETg(y).

Itis dear from thehypothesesthatfor all y

e Y,

the linearfunctional

(y

is inthedualspaceofh,and hasnormless thanor

equa

to

M

IITII.

Thus

Qy(g)

[g,a]

for some a

e h,

with

Ilall

<_

M.

It

followsthatfor eachorthonormalset

{ei}

ofelements of

h,

andfor each

finite

N,

(ziN=l

IETei(Y)l

2)

1/2

e

C(Y)

asafunctionof y, withsupremumnorm lessthan orequal

N

2)112

EIGENFUNCTION APPROXIMATIONS FOR CONTINUOUS SPECTRUM OPERATORS

absolute values. It follows immediately that for almost every x in

X

with respect to p,

(i=l

[Tei

(x)12)1/2

-<

M,

as we desiredtoshow.

Lemma

1.6. Letn be apositive integer.

Assume

thehypotheses

of

Estimate 1.3. Then the

followinghold:

i)

B isahomeomorphism

from

W

onto

W,

and

B’

isahomeomorphism

from

W’

onto

W’.

ii)

There existsaconstant Ksuchthat

.for

any

e

domain

H

N,

II(B’)/t]](R)

_<

K(11112

+

]]H

N]]2

).

iii)

Let

{(i)}ik=l

be anypairwisedisjointcollection

of

Borel subsets

of

R;

letK be as in

ii)

above.

Suppose

that e is in the domain

of

H

N,

and that

{0i}

is any collection

of

elements

of

W

such that 0

i]12

<-

1. Then

i=1

]

(i)

UeB0i

dae

(A)

-<

where

[[el[h

2

Ile][

+

[IHNe[[,

r

is the union

of

the supports

of

the

functions

0_i, and

l(r)

isthe characteristic

function

of

r.

Proof."

Toprove the first conclusion, we note that the mapping B is a 1-1 mapping from

Wonto itself. This implies that B is a homeomorphism,byaresult ofDieudonne and Schwartz

(see

the discussion onpage 124 of

[7]).

However,

it is now easy to prove this result from later

work onwebs; for completenesswegive the proof. It is clear that thegraphofB-1isclosed.

By

Theorem2, page 158,

[7],

we seethat if

E

is a Frechet space, and

F

isa separatedconvex space

with a completing

web,

then mappings from

E

onto F with

(sequentially)

closed graph are

continuous. Itis dearfrom the definitionofacompatible web that aFreehet space orthe strict

inductivelimit of Freehet spaceshas a compatible

web,

whichiscompleting by Lemma 1, p. 156,

[7].

Hence,

Whas acompletingweb. Toshow the continuity ofB

-1,

we needonlyshow that the

restrictionof

B

-1to each spaceV_niscontinuous, sincefor an open set

U,

B-I(u)

istheunionof

its intersection witheach

Vn,

andsinceasetisopeninWifandonlyifitsintersection witheach

V

nis open.

But

eachVnis a Frechet space, andit isobvious from the continuityof

B

that the

graph oftherestrictionof

B

-1 to eachV

nis sequentially closed.

Ther.efore

B

-1

iscontinuous, as we desired toshow. That

B’

is a homeomorphismis immediate.

The second conclusion is aconsequence of the dosedgraphtheorem, becausethe mapping

T

taking the domain of H

N,

with graph norm, into

L(R)(X,p)

has closed graph, where

T

{B’}/f.

To

see this, suppose that

Cn

is a Cauchy sequence ofelementsof the domainof H

N,

in graph norm, and that

Ten

is Cauchy in

L(R)(X,p).

Then

Cn

convergesin

L2(X,#

to an

element ofthe domainof

H

N.

We

mustshow that

Ten

converges

to

T.

Since

Cn

convergesto

in

L2(X,p),

then

Cn

converges

to in

W’,

where we haveembedded

L2(X,p

into

W’

using

v)

of Assumption 1.2 by mapping g to the linear functional

Fg

such that

Fg()

[,g].

Since

B’

is continuous from W’ into

W’,

then

B’n

convergesin W’to

B’.

But byhypothesis,

(B’n)/f

is

Cauchyin

L(R)(X,p)

andtherefore convergestoanelement 0 of

L(R)(X,p).

The precedingargument, using

v)

ofAssumption 1.2for

Ll(X,p),

shows that convergencein

L(R)(X,p)

impliesconvergencein

8 R.M. KAUFFMAN

fF()

F(f),

because the transpose ofa continuous map from W intoW is a continuous map

from W’intoW’. But wehaveembedded

L2(X,p

into W’ using an embedding mapEsuch that

E(f)

fE().

Thus

B’n

convergesto f0inW’. Thus

B’

f0in

W’. Hence

B’(x)

fx)

for almosteveryxinX withrespectto p. Therefore

(B’)/f

,

aswedesiredto show. The second

assertionisproved.

Weprove the third assertion.

Note

that the second conclusion ofthe lemma guarantees

that for any inthe domain of

H

N,

II(B’)/fll(R)

<_

2K(IIII

+

HNII)

1/2.

We may therefore

use

2K

as the constant M in Lemma 1.5. Define b by

bi(A

IU,,e(B0i)I/Ue(B0

i)

if

Ue(BOi)

# 0; define

bi(A

to be 0otherwise. Notethat each b is measurable with respect to a

e.

,

Use r to denote the complex conjugate of thefunction r. Let

R((i))

denote the characteristic

function of

(i),

and define g S

e by

Ue(g

ii=likt((i)).

By Lemma

1.5, we see that if

e

(P((i))g)/llP((i))g][

_h,then

iil

IB’ei

(x)[2

<-

2K2f2

foralmosteveryx. Thus

i--1

.t

((i)

Ue(B0i)

dae(A)

i_l/’((i)bi(,)Ue(B0i)dae(A)

i=l/"

{Ue(BOi)}Ue(P(((i))g}*(A)dae(A)=

ii=l[B#i,P((i))g]

i=lir

0i(x){B’P(((i))g}*(x)dp(x)<-

i=l

Ilz(r)s’P((i))gll2

ii

=lllP((i))gllhlla(F)B’e’ll2

_<

(i=lllP(((i))gll)l/2(zk.

=llla(F)B,eill.)l/2.2

But

i=l

[[R(F)B’ei[[

i=l/’F

[B’ei

[2dp

-

2K2J’l

f2dp"

Hence

ii=l]((i)[Ue(BOi)ldae(A)<-q2K(ii=lllP(((i))gll)l/2llR(r)fl[2-<

2Kllellhlll(F)fll

2.

Thethird assertionfollows immediatdy; the lemmaisproved.

Corollary 1.7.

Assume

thehypotheses

of

Estimate 1.3. Usethe notation

of

Definition

1.4.

Suppose

that e is in the domain

of

H

N.

Then

for

every e

>

0 there ezists an

M

>

0 such that

ae(AM)

<

e, where

A

_M

{A

II\A0[

3

e

Swith

IZA,e(B)I

_>

MI1112}.

Let us write the countable set S as

{i}"

Let

(iM

{’

IZ,,e(Bi)l

->

Proof."

MI1i112.

Let

0

i/ri,

wherer is arational number such that

I1i112

-<

ri

<-

211&i112-

For any fixed

M,

let

(1M

and

:7j

(jM

\

U[

(rM"

Then foranyk,

:71

i=l

]:7i

IUe(B0i)ldae(’X)

->

(M/2)ae(Uik=l:Ti)"

By

Lemma 1.6,we seethat

%(u.

k

,=1:7i)-

<

(/M)72Kll’llg.(IIP(Ui=iri)elI

+

IIrlNp(u=ir/i)eII)

1/9.

Thecorollaryis proved.

Theorem1.8.

Assume

thehypotheses

of

Estimate 1.3.

Suppose

that e is inthe domain

of

H

N,

andthat

IIh

is as inAssertion

iii)

of

Lemma1.6. Then thereexists asubset

of

suchthat

ae($

\

()

0 and such that

for

every E

(,

thereezists aunique element

Z,,e

in

W’

such that

Z,,e

isnot thezero

functional

and

ZA,

_e agreeswith

ZA,

eonS. Furthermore,

if

FA,e

is

defined

on by

EIGENFUNCTION APPROXIMATIONS FOR CONTINUOUS SPECTRUM OPERATORS 9

i)

H’F,

_e

F,

whereH’ isthe conjugate

of

the restriction

of

H to

W;

furthermore,

for

any 0e

W,

F/3,e

(0)

Ue(0)(/3)

.for

almost every

13

withrespectto

ae;

ii)

S’F,,e

{

L2(X,p);

iii)

if a()=

B’F

then a is a measurable

nction

with respect to a

e

from

into

L2(X,p)

inthesensethat e

>

0 thereexists acompactset

F

such that

ae(

\ F)

<

e and such that therestriction

of

ato

r

is a continuous

function

from

F

into

L2(X,p);

iv)

if3

isaBorelsubset

of

X

and A isaBorel subset

of

,

then

]A

[[:(/3)a(’)ll2dae

<

q’K[IP(A)ellhll2:(/3)f[12’

where

y,(3)

isthe characteristic

nction

of

3.

Proof"

B’

is a homeomorphism from

W’

onto

W’.

By

the preceding lemma, for almost

every the functional

Z,,e

hasa unique extension

Z,,e

toW. Thereexists aunique element of

L2(X,p

which agrees with

Z,

_e on

W;

denote this also by

Z,

e.

For any element of

S’,

Z,e() B’F2,e() F2,e(B)

foralmostevery

,

withrespect toa

e.

But

also

Z2,e()

Ue(B)()

0

for almostevery ).

If6is the set of such that

Z,

e is the zero functional, then for every in

W,

Ue()

vanishes on 6.

By

continuity ofU

e from

L2(X,p

into

L2(ae)

and since W is dense in

L2,

it follows that U

e(e

vanishes on 6. But this functionis identically equal to 1 almost everywhere

withrespecttoa

e.

Hence

ae(6)

0. Thus

Z,k,e

isnon-zerofor almost every

.

By

the spectral theorem for self-adjoint operators in a Hilbert space, if

B

0, we see

that

F,e(H0

Z,e(B-1H0

Ue(H0)(,

,Ue(0)(,

exceptfor afixed set ofmeasure 0 with

respect to a_{e, which} is the union of exceptional sets for each in the countable set S’. Since

F,,e

E

W’,

andsincetherangeof therestrictionofBtoS’isdenseintherangeofB andtherefore

densein

W,

the first conclusion follows immediately. The second assertion was proved at the outset.

We provethethird assertion. Notethat if

Be

e

S,

,e(B)

is measurable bydefinition.

Extending by continuity, for almost every ,k, to

L2(X,p),

wesee that’

[,B’F2,e]

is ameasurable function of

,

for every fi

L2(X,p

).

By

proposition 8.15.2, p. 574, Edwards

[3],

the conclusion

follows.

Notethat

[B’F,,e,

F,,e(B)

Ue(B)

almost everywhere.

By

passingtothelimit in

Lemma

1.6we obtainthatfor any openset

F

in

X,

andanypartition

{f(i)}

of

A,

and any setof

elements

i

e

L2(X,p),

each supportedin

F,

such that

I1ill

1,

i=l’t

A

I[n’fA,e’i] dae

-<

Now use assertion

iii)

to select for every e

>

0 a compact set R of

A

such that

ae(A\e)

<

and such that the restriction of

B’FA,

_eto R is a continuous functionfrom R into

L2(X,p

).

Itfollowsthat,if

]]2,r

denotes thenormof

L2(r,p),

i0 R.M. KAUFFMAN Fromthe monotoneconvergence theorem,itfollows that

A

llB’FA,ell2,rdre

<-

Kllx(r)fll211P(A)ellh

Selecting acountable decreasingchainof opensets

r

with intersectionessentially equalto

/,

and using the monotone convergencetheorem again, we easily complete the proof of

iv),

and hencetheproof ofTheorem1.8 iscompleted.

2. SOME

REPRESENTATIVE EXAMPLES

In

thissection, we take a look at some representativeexamplestomotivatethetheory.

Ezsmile

2.1. Let

X

[Rn and p be Lebesgue measure.

Let

W denote

C([Rn).

It

is

well-known

(see

p.

75,

[7])

that W satisfies the hypotheses above.

Suppose

that

H

is a

self-adjoint operator in

L2({

n)

such that

H

-

for all in the domain of

H,

where

-

is a partial differential expression withC(R)

coefficients. Supposefurther thatforsome positiveinteger

N,

allelements ofthe domainof

H

Nliein

L(R)(iRn).

Let

abeany

bounded,

L2,

positive elementof

C(R)(n).

Let

B

be multiplicationby

;

then

B’

is also multiplicationby

.

Letfin Estimate 1.3

bew. Theorem 1.8 then yieldsthat,for anyeinthe domainof

H

N,

theeigenfunction

FA,e

has the

propertythat

FA,e

is an

L

₂function, whichhas the properties of

B’FA,e

in this theorem.

Here

rFA,e

AFA,e

inthe senseofdistributions.

Ezsmple2.2. Supposethat

H

is aself-adjointoperatorin

L2(iRn

such that Hf

r/f

forall f

inthedomainof

H,

wherer/is a partial differential expression withC(R)

coefficients such that each derivative ofany coefficientof

r/has

at mostpolynomial growthatinfinity. Supposethat H has the additional property that for any positive integer j, thereexists a positive integer

N(j)

such that the domainof

HN(j)

is contained inthe Sobolev space

HJ(n).

Let

-

denote the differential

expressiondefinedby

-=

ii=l

2/bx-

1. Let Bbethe operatoronWdefinedby

B

(Ixl

2

+

l)-(n+)/4,

wherer islarge enoughsuch that for any gsuch that g6

H2r(iRn),

itfollows thatg

e

L(R)(n).

Let

Wbe the space of rapidlydecreasingfunctions on

n.

By

using the

Fourier

transform,

we seethat B satisfies thehypotheses ofEstimate1.3.

W

satisfiesAssumption

1.2 since it is aFrechet space. Direct calculation shows that

B’(8)/(Ix12+l)

"-(n+)/4)

E

L(R)(iR

n)

for all 8in thedomain of

H

N(j),

where is large enough that for 8i

HJ([Rn),

DaB

e

L(R)(

n)

for

lal

_<

2r.

By

Theorem 1.8, wesee that

rr{(Ixl

2

+

1)-(n+)/4FA,e

}

e

L2(n).

Since

FA,

_eis in

W’,

the space of tempered distributions, we see from the ordinary Fourier transform that

(II

+

l)-(n+0/4F,

_e

e

L(R)(n).

Ezample2.3.

Let

H

be any self-adjoint operatorin

L2(iRn

such that H8

r/#

for all 8in

the domain of

H,

where

r/is

apartial differentialexpression with

C

(R)

coefficients,

such that each

derivativeof each coefficient of

r/has

at most polynomialgrowthat infinity. Let Wbe the space

ofrapidlydecreasingfunctions.

Suppose

thatWiscontainedinthedomainof H. Lete

L2(iRn

).

w,

( i=1

1)-m((l l

taken in thespace

W’,

and where

M

is such that

(Ixl

2

+

1)

TM

is in

L2(IRn

).

Then

B

satisfies Assumption1.2 andEstimate1.3,with f

(Ixl

2

+

1)-7,

withN 0, since

EIGENFUNCTION APPROXIMATIONS FOR CONTINUOUS SPECTRUM OPERATORS II

(ii=l

2/x-

1)-M0

E

L(R)

(zn)

for all 0E

L2(X,p

).

Theorem1.8 shows

that,

for

most

every %

2

)-NFA,e

withrpect to

e’

F%,e

W’and

(Ix[

2

+

1)-?(=1/

1

L2(n

).

3.

APPROXIMATION

In

ts

sectionwestudythe

mn

problemof thepaper.

spon

3.1. Let

Y

be a subspace of

L2(X,

wchis

so

aBanach spacewith norm

[[y,

andwhichh the properties that

a)

Wisadensesubspace of

Y,

b)

thereestsaconstt

J

such that

J[[[[y

[[B-l[[2

for 1

W,

and

c)

the

intion

om

Y

into

L2(X,)is

continuous.

Let Z

denote

{F

W’[

B’F

L2(X,)};

if

F

Z,

let

[[F[[

Z

[[B’F[[

_2. Let

A

denote

thecontinuous extensionof

B

-1as operator

om

Y

into

L2(X,

).

Rr: We

now

ve

the deflation of agonzation, d introduce a

sctr

meure wch we denoteby

e"

Sincethepropertiesof

e

everyimporterfor o theory,itis usefto notethat byan

ementy

ccationitfollowsthat the deflationof

e

does notdepend upone,

butonly uponS

e.

In

otherwords,ifS

e

Sp

then

e

f"

It should

so

be remarked that the

following deflation has been made

qte

detled

cause

it sms usef forlater appcation to statethe appromation properties we obtncompletely.

Dtion3.2. Supposethe

hotheses

ofAssumption3,1, Assumption 1.2 d Estimate

1.3

hold,

d that e domn

(HN).

Let

e

be the positive meure on definedbythe relation

[[B’F,e[[de,

where

F,

e is in Threm 1.8.

Let Q

r(H),

where r is a unded

d

_e

continuous fction

om

the

sctrum

of the restriction of

H

to S_einto

K.

Let be a

Bor

subset of

.

We say that

P(A)P(Se)

Q

is

doiMe

in

B(Y,Z)

th

resct

to

H

d e if

B’P(A)P(Se)Qg

L2(X,

for g

Y,

d

a)

for every

>

0 there

ests

a positiveinteger k and a fite

sint

faly

{Ai}=lof

subsetsofisuchthat

e

(Ai)

is fiMtefor every dsuch that there estsaset of

reM

humors

{Ai}=

1th

A

Ai0A

d th thepropertythat,inting

P(A)P(Se)Q0

cocy

into

W’,

II{P()P(Se)Q-

=1

e(AinA)7(Ai)RAi,e}(O)l[Z

[[0l[y

for

O

e

Y,

where

b)

A

is inthe complement ofthe ceptionM set of Theorem1.8,sothatinptic

FAi,e

is inThrem 1.8 d

B’FAi,e

L2(X,p

th

]]B’FA,e]]2

#

0, andwhere

c)

RAi,e($

B

GAi,e(A)GAi,e

for any

Y,

where

GAi,e

FAi,e/]]B’FAi,e]2,

d

where

B

’GA,e

denotesthe complex conjugate ofthefunctionM

B’GA,

e.

The complex conjugate appes

agMn,

cause

we e

worMng

in

W’. Note

that wle the points

A

dependon

A,

the

number k and the sets

A

do

not;

these depend

oy

upon

.

Note

Mso

that

RAi,e

agrs with

G

Ai,e($)GAi,e

on

W,

d that byby,thesis

b)

of Assumption 3.1, togetherth thefact that

B’GAi,e

L2(X,),

itfollowsthat

=

R. M. KAU F FMAN

Let Mbe afamily of bounded continuousfunctions fromthespectrum of the restrictionof

H to S

e into t:. Let

QM

(r(H)lr

E

M}.

We say that

P(A)P(Se)QM

is simultaneously

diagonalizablein

B(Y,Z)

withrespect to

H

and e iffor every

>

0thereexist

A

and h as in

a),

b)

and

c)

above such that ₁₌₁

(0)llz

_<

,ll011y

for all

0E Yand allr

c

M.

Theorem 3.3.

Suppose

that the hypotheses

of

Assumptions 3.1 and I.

hold,

and that

Estimate 1.3holds, ande domain

H

N.

Let

G,,e

F,,e/IIB’F,,ell

2. Then Assertion

i)

below

implies Assertion

ii),

which inturn implies Assertion

iii).

i)

A

is aBorel subset

of

such that

{B’G,,e

,

c

A}

isprecompactin

L2(X,p

).

ii)

] A

IIB’F,,elldae(’)

<

(R)"

(an

elementarycomputationusing thespectraltheorem shows

that

/’A

IIB’F,ell2d%

!ZX

IIB’F,gll2dag_

if

gisanother cyclicvector

for

thesubspace

Se.)

iii)

Let

Q

r(H),

where r is a bounded continuous

function

from

the spectrum

of

the

restriction

of

H to S

e into f. Then

P(A)P(Se)

Q

is diagonalizable with respect to H and e in

B,(Y,Z).

IfM

is aset

of

boundedcontinuous

functions

from

thespectrum

of

therestriction

of

Hto

S_e into f which is uniformly bounded and equicontinuous on

A,

then

P(A)P(Se)Q

M is

simultaneously diagonalizablewithrespectto

H

ande in

B(Y,Z).

Proof:

Wenote that Theorem 1.8guaranteesthat B’F

into

L2(X,p

with respect to

ae,

in the sense that forevery e

>

0 thereexists a compact set

K

such that

ae(\K

<

and such that the restrictionof

into

L2(X,p

).

Weshow thatAssertion

i)

impliesAssertion

ii).

In

fact,

if Assertion

i)

holds, thereexistsa

finite set

{i}ik=l

of points of A such that for all

A,

there exists a

’i

such that

IIB’G,,e-B’G,i,ell2

<

/2.

Select a set

{i}ik=x

ofelementsofWsuch that

[B’G,i,e,i]

>

3/4,

and such that

I1i112

-

Let

A

t’kl=l

Ai,

wherefor

,

E

A

_i,

IIB’G,,e-B’G,,elI2.

<_

/2.

We mayassumewithout loss of generalitythat thesets

A

are disjoint. It follows thatfor all

,

C

Ai,

I[B’GA,e,i]I

>

1/4.

Hence

I[B’FA,e’i]I

>

IIB’F,,elI2/4"

But

[i,B’FA,e]

Ue(Bi)(A

).

Hence

I

A

IIB’FA,elldae(A

<

16ii=l/A

IUe(Bi)]2dae(A).

Since

Si

W,

theintegral onthe fight

isfinitebythespectraltheorem.

To complete the proofof the theorem, we need a lemma, which has some independent

interest.

Lemma3.4.

Assume

that Assertion

ii)

of

Theorem3.3holds.

Define

#e

on

A

asabove. Let

7(A)

be

defined

on

A

by

OA)=

GA,

_e

FA,e/IIB’FA,ell

₂

for

all

A

ina subset

A

0

of

A

such that

ae(A\A0)

0.

(This

definition

makessensebyTheorem

1.8).

Then thefollowingaretrue.

i)

Forany

fl

e

LI(A,#e

),

37

isascalarwise integrable

function

from

A

intoW’ withrespect

to

%,

where by

definition

this meansthat

for

any

c

W,

37()

E

LI(A,#e).

In

particular, this is

EIGENFUNCTION APPROXIMATIONS FOB CONTINUOUS SPECTRUM OPERATORS 13

ii)

I.f

F(/)/s defined

.for

every E L

l(#e)

and W by

F(/)()

.f

A

"(A)G

A,e(

)de

(A),

then

F(/)

(

W’.

iii)

For

every/

e

LI(A,%),

’?

ascalable

inferable

nction #ore

A

into

L2(X,p

th

t

to

%,

’

(

)’

,e"

iv)

ff

()

is

defined

for

eve

e

LI(A,%)

and

e

W by

(Z)()

la

(A)B’G,e()d%(%),

en

@()e

L2(X,p

),

where

L2(X,p

,

identified

th iU canonicalembeddingintoW’. fuheore,

]]@()]]2 lfl]]l"

v)

gZ

e

Ll(a,%),

B’r(Z)= ().

)

g0

e

L2(X,p

),

and

Ue(0)/I]B’FA,e]]2,

en

e

LI(A,%)

and

()

B’P(A)P(Se)0

where

B’P(A)P(Se)0

identified

thi canonicalembeddingintoW’.

In

paicular,

B’P(a)P(Se)0 L2(X,o

).

Proof"

Part

iii)

follows

om

the finiteness of the measure

e’

together with the

boundedness of

B’G,

_ein

L2(X,p

).

Since Bis 1-1 donto, part

i)

follows well. Since

e

is

fiteon

A,

L(A,e)

c

Ll(e).

We

show that

@()

W’. By

Threm 1.8, we knowthat isa measurable function from

a

into

L2(X,p

).

Furthermore,

@()()

[1][2,

so we s that

()

agreesonWwith a unique element of

L2(X,p)

defined

a

the

esz

representation threm for Hilbert spaces by the relationsp

@()(0)=

la

()[0,B’G,e]d%()

for 0e

L2(X,p).

Part

iv)

is proved. Since

(B’)

-1is acontinuous fine mapping

om L2(X,p

into

W’,

(canse

the injection of

L2(X,p

intoW’ iscontinuous, d

(B’)

-1isacontinuous line trsformationin

W’),

where we once

agn

identify

L2(X,p

th its

canocM

emdnginto

W’,

we may use Theorem8.14.5,p.562,

[4]

toobtnpart

ii)

andpt

v).

We provept

vi).

If fi

W,

itfonows from part

iv)

that

[,@()]

IA

()[’n’G,e]d%()"

Also,

[,S’G,e] B’G,e() G,e(S)

Ue(S)/[[S’F:e[[2.

Hence

]A

(’X)[’B’GA,e]d#e

(’x)

]A

(Ue(B)/IIB’F,x,elI2){(U

_e

0)/llB’G,x,ell2}d/e

(A)"

But

[[B’F,x,e[[2dae

so the integral on the right becomes

-fA (Ue(B))(Ue 0)dae(’X),

which

d#

_e

equals

[B,

by the spectral theorem; this theorem also guarantees that the integrand is in

Ll(ae)

and also that

/

(

L2(e

).

Since

e

is finite on

A,

it follows that

/

(

LI(A,e).

But

if

a=

P(A)P(Se)0

[Be,

a]

a(B)= B’a(),

again embedding

L2(X,p

canonically into

W’.

We

havethereforeseenthat

B’a

(/),

as we desired to show. The lemmaisproved.

Wenowshow that Assertion

ii)

implies Assertion

iii)

ofthe theorem. For anypositive real number

,

select a compact subset

K

of

A

such that h is a continuous function from

K

into

L2(X,p),

and such that

#e(A\K)

<

/. Notethat

{h(A)]

A

e

K}

is a compact subset of

L2(X,p

).

14 R.M. KAUFFMAN

a(j)

h-l(j)).

Select afinite relativelyopencover

{R(n)}

of

K

whichissubordinate to

{a(j)},

and whichhasthe property that

]r(A)-r(/J)l

<

forall

A and/J

in

R(n).

Let

((j)

R(j)\Ui<jR(i

).

Let

bj

#e((j)).

Select one

Aj

from each set

((j)

and let

Qo

j=l-(Aj)bjRj,e

using the

notationofDefinition 3.2. Notethatfor E

W,

since

IIB’GA,e]I2

1,

GA,e

()1

B’GA,e

(B-I#)I

I[

B’GA,e

,B-I]I

-< II

B-1

II211B’GA,e]I2

<

J}}

#lly-Using

Lemma

3.4, parts

iv),

v)

and

vi),

togetherwith Proposition 8.14.6,p. 562, Edwards

[4],

used on the space

L2(X,p),

we see that for any E

W

such that

{{{{y

1, the function mapping

A

to

(A)A,e()GA,e

is a scalarwiseintegrablefunction from into

W’

withrespect to

e’

and

IIB’{P(Se)P(A)Q

Q0}ll2

IIB’{j$

(j)

((2)A,e()G,

e

7(Aj)gAj,e()Gj)dPe(A)

+

]A\K

()gA,e()GA,ed#e(A)}ll2

IIB’{ZjliCj)(Y()- ’(Aj))g2,e()G2,ed#e(A)+

jT(Aj).t

(j)

(gA,e()

gAj,e())GA,ed#e(A)

+

Ej(Aj).

(j)

Aj,e

()(GA,

e

GAj,e)d#e(A

+

/A\K

)-(2)gA,e()GA,ed#A,e}ll2

-<

$

Zj./(j)

Jd#e(A

+

Ilrll(R)jI(j)

JIIB’(G2,

e

GAj,e)ll2d#e()

+

Ilrll(R)jl

((j)

JIIS’(G,e-

GAj,e)ll2d#e()

+

Ilrll(R)Jl\Kd#e()

<_

J#e(a)+ JIIrll(R)#e

(h)

+

JIIrll(R)#e(a)

+

JIIrll(R)-Wehave seenthatforany

e W

such that

lly

1,

IIB’{P(Se)P(A)Q--Q0}112

_<

5{J#e(A)(l+211rll(R))

+

JIIrll(R)}.

Wemust extend thisestimatetoallof Y.

Weexaminetheoperators

B’P(Se)P(A)Q

and

B’P(Se)P(A)Q

_0.

Note

that by

Lemma

3.4, for any 0in

L2(X,p)

xP(/3)=

B’P(A)P(Se)0

where/=

Ue(0)/[IB’FA,elI2.

Again by Lemma 3.4,

()

e

L2(X,p),

and

11()ll2-<

IIll-

However,

since # is finite on

A,

it follows that

11()ll2-<

(e())l/211ll2

by the Schwartz inequality. But the spectra] theorem, and the definition of

#e’

show that

II(/X)ll2,e--IIP(/X))(Se)0112,

p.

If 0-

Q,

with {

Y,

and if we recall thatby Assumption3.1 theidentitymapping

Ey

from

Y

into

L2(X,p)

is continuous,wesee frompart

vi)

ofLemma3.4 that

B’P(A)P(Se)

Q

isacontinuousmapping from

Y

into

L2(X,p

).

and that

RAi,e(

B

GAi,e(A)GAi,e

for any

Recall that

Q0

ii=l

#e(i)(Ai)RAi,e’

EIGENFUNCTION APPROXIMATIONS FOR CONTINUOUS SPECTRUM OPERATORS 15

Observe thatwehave selected

A

insuchaway that

B’FAi,e

E

L2(X,p).

Furthermore,

B’GA.

(A)

[A,B’GAi,e

].

Since

A

is continuousfromYinto

L2(X,p),

it follows that

Q0

is a l,e

bounded linear transformation from

Y

into

L2(X,p).

Assertion

iii)

of Theorem 3.3 follows by

continuity,sinceWisdenseinY. The theoremis proved.

4.

A

SIMPLE

EXAMPLE

In

thissection, toplacethe resultsinperspective, westudythesituationofasecond order ordinary differential operator in

L2([0,(R))).

We show that in considerable generality in the second-order ode

case,

thereexistunbounded sets

A

and natural spaces Y and

Z

such that

P(A)

is diagonalizable

(and

therefore

compact)

in

B(Y,Z),

but such that the injectionfrom

Y

into

Z

is

not compact. Hence in this situationthe compactness of

P(A)

is a consequence, rather than a

hypothesis,forourtheory. Forpidentically1and

xq(x)

E

L1,

sets

A

havethisproperty forthe

Y

and

Z

ofthissectionif

1//

LI(AI(0,(R)))

asveryrecent resultsof D. B.Hintonand the author

[4]

show, using the results ofthis paper.

A

number oflemmas are proved in this section, which

seem fairly obvious, such as the existence of a cyclic vector. The proofs and statements are

included because the authoris unable to find themintheliterature, in thecontinuous spectrum

ase.

Definition4.1. Let rbe givenby

re

-(pC’)’

+

q

for allsufficiently differentiable on

[0,(R)),

where p and q areC(R)

functionsfrom

[0,(R))

into

I,

and where pispositive and boundedaway from0and qisbounded below.

Let

H

be the Friedrichs extensionin

L2[0,(R)

oftherestrictionofr

to

C(0,(R)).

It

is clear that any element ofthe domainof

H

is actually in

L(R)[0,(R)),

since the

hypotheses guarantee that for such a

,

’

is in

L

₂

L2[0,(R)

).

Throughout this section let w be any positive bounded C(R)

element of

L2;

let Z

w denote

{F:

wF

L2},

and

Yw

denote

Remark 4.P.. If W

C(0,(R))

and

B

is the operator of multiplication by

u,

then Assumption 1.2, Estimate1.3, and Assumption 3.1 holdfortheabove

H,

with f

w, Y

Yw

and

Z

Zw.

In

particular, Theorem3.3 andLemma3.4 hold. Weshow that thereis andement eof

Y

such that S

e

L

2. This requires another functionalanalyticresult.

Theorem 4.3.

Suppose

tha Assumptions 1.and .1 andEstimate 1.3 hold. Then

for

any

e

e

D(H

N)

and almost every) withrespect to

ae,

thereexists adecreasing tower

{

An}

of

compact setswiththefollowingproperties:

a)

A

_n

[- l/n,A

+

l/n];

b)

ae(An)

>

0;

c)

the mappingG

,e

iscontinuous

from

A

ninto

Z;

d)

P(an)e/e(an)

convergesin

Z

to

G,e/llS’f,,ell

_2.

Proof."

Weshowedin Theorem 1.8 that the mapping 7:

,-G,

_eis ameasurable function

16 R.M. KAUF FMAN

A

n

[

1/n,

+

l/n]

N

Kj.

If

e(An)

0for all n, and E

A

nfor all n, then letting

n

denote

thecharacteristicfunctionof

An,

Lemma3.4, part

vi)

guaranteesthat

r(n(S)/[[B’Fs,e[[2

-P(An)e

since

Ue(e

_:1 and since

[[B’F,eI[2

is bounded away from 0 on

An,

becauseit is continuous and non-vanishing.

But

it is clearfrom

vi)

of Lemma3.4 together

withthe continuity of7that conclusion

d)

ofthe theorem holds.

However,

the set of all such

thatfor everyset

Kj

containing thereis an n such that

e([

1/n,

+

l/n]

N

Kj)

0 is aset of measure0 withrespect to a

e.

Thetheoremisproved.

Theorem 4.4.

Let

H be as in

Definition

.1.

Let

W and

B

satisfy the hypotheses

of

Assumptions 1.2 and 3.I and Estimate 1.3; assume

further

that

C(0,(R))

is algebraically and

topologicallycontainedin

W,

where

C(0,(R))

isgiven the usual inductive limittopolo.

(Recall

that

a//

E

C(0,(R))

vanish in a neighborhood

o/

0).

Supposee Domain

(H).

Then

FA,

_e

e

C(R)[0,(R))

and

rFA,e

AFA,e

for

every

A

inthe set

of

Theorem1.8, and

FA,e(0)

0

for

almost every

A

in withrespecttoa

e.

Proof."

Let

A e

.

Then

H’FA,

e

AFA,e,

by Theorem 1.8.

Hence,

for any # in

W,

F

A,e(rO

AFA,e(0).

In

particular this is truefor 0 in

C(1,(R)),

so that

rFA,e

AFA,e

in the

spaceof distributions. But anydistributional solution to

rFA,

_e

AFA,

_eis in

C(R)[0,(R)).

Weneed

onlyshowthat

FA,e(0

0for almost every

A

withrespect to a

e.

For this, we may choose the

spaces W and Y and the map B any way we wish. Choose W to be

C(0,(R))

and B to be multiplicationby

w,

withf w.

By

Theorem 4.2 we may construct a sequence A

n ofcompact

sets contained in

[A+I/n,A-I/n]

for almostevery

A

such that

n

P(Ane)/#e(An

converges in

za

to

G,,e/llwF,,ell

2. But since

A

nC

[,-l/n,A-I-1/n],

it follows that

rn

converges

in

Z5

to

,G,,e/llwF,,ell

2.

Hence,

if

n

aCn’

we see that

n

converges in

L

2 to

3,e/llF,,ell

2 and

ar(n/

converges in L

2 to

,3,,e/llwF,,ell

2. Writing out the terms of the differential expression

B

such that

/0=-o-r(0/a),

we see that

n

converges uniformly on compacta to

G,,e/llF,,ell2;

in particular this is truein a neighborhood of0. It follows immediately that

GA,e(0

0, as we desired toprove.

Theorem4.5. Let

H

be as in

Definition

4.1.

Then there ezists ane in the domain

of

H

such thatS

e

L

2.

Proof.

ChooseW

C(0,(R)),

and choose

B

inEstimate 1.3 to be multiplicationby aand f

.

Choose

Z

Za.

From the spectral theorem, there exists an e

L

₂ such that for any gE

L2,

ag

isabsolutelycontinuous withrespect tor

e.

Without lossof generality,emay betaken

in the domain of H. Weshow that S

e

L

2. Ifnot, thereexists a non-trivial element gofthe domainofHsuch that

Sg

S

e.

By

the previous theorem, for almost every

A

withrespect to

Se,

F,,e(0

0. Similarly, for almost every

,

with respect to

Sg, FA,g(0

0. Let the sequence

Cn

C(0,(R))

be such that

I1n112

1 and also such that

Cn

converges

to g in L

EIGENFUNCTION APPROXIMATIONS FOR CONTINUOUS SPECTRUM OPERATORS 17

subsequence, which we also denoteby

n’

we see that

Ue(n)(A

converges

to 0foralmostevery

withrespect to a

e and

Ug(n)(A

converges

to 1 for almost every with respect to

ag.

Hence,

since

g

isabsolutely continuouswithrespectto

e’

we see thatforalmostevery

A

withrespectto

g, F,e(n

converges to0 and

FA,g(n)

converges to1.

But

FA,

_emust be a multipleof

F%,g

for almost every % with respect to

g,

sinceboth vanish at0. Thisisacontradiction; the theorem

isproved.

Remark: Wenowprovea negativeresult, showing that somehypotheses arenecessary in

order to diagonalize on a Borel set

A.

The first operator one might wish to diagonalize is the identity operator; this gives a discrete approximation to aninverse Fouriertransform.

Theorem 4.6. Let H be as in

Definition

g.1. Suppose that

P(A)

is diagonalizable in

B(Y

_w

,Zw)

withrespect to H. Then

p(A)

is acompact operator

from

Y

_winto

Z

w.

In

particular, the identityoperator

P()

isnot diagonalizablein

B(Y

w

,Zw)

withrespect to H.

Remark: Sincethe embeddingof

Y

_winto

L

₂is continuous, asistheembedding of

L

₂into

Zw,

itfollowsthat

P(A)

is in

B(Y

_w

,Zw).

Since the previoustheorem showedthat there exists an dement eofthe domainof

H

suchthatS

e

L2,

wedo notneedtoconsider

P(Se).

Proof"

Operatorswithfinite dimensionalrangearecompact,asarelimits inoperator norm of such operators. The first conclusion isthereforeimmediate. The embedding of

Yw

intoZ_wis

dearlynot compact,since on theinterval

[0,1]

the normof

Yw

isequivalent to thatof

Z

_wand to the normof

L2([0,1]).

Theorem 4.7.

Let H

be asin

Definition

4.1; let e beany element

o/the

domainof

H

such thatS

e

L

2. Then

a)

for

anyboundedcontinuous

function

r: spectrum

(H)

P(A)r(H)

isdiagonalizablein

B(Y

_w

,Zw)

withrespect to

H;

b)

if

the essentialspectrum

of

H

isnotabounded

set,

thereexistsubsets A

of

thespectrum

d#

_e

dae)

and such that

for

all

N,

of

H

such that

%(5)

is

finite (where

ae(A\[-N,N])

>

0.

In

particular,

for

such

A

and any bounded continuous

function r(H)

of

H,

P(A)r(H)

is diagonalizable with respect

o

H

in

B(Y

_w

,Zw),

although

A

is not an essentially bounded setwithrespect toa

e.

Remark: The first conclusion of the theorem does not state that

#e

is finite on bounded

sets. Thisquestionisadifficultone,whichwedo not address here.

Proof:

The first assertionwillbe proved in more generalityin Theorem 5.5 of the next

section. If the second assertionofthe theoremis

false,

then the finitenessof

#e

(A)

implies that for some

N,

ae(A\[-S,S])=

0. This implies that if a

e([S+i,N+i+l)) {ai:

a

>

0}

is

boundedaway from 0in

(0,(R)).

Since theoperator

H

is

unbounded,

if

F

{i:

a

>

0},

then

F

is

glb

{#e(J)" #e(J)

>

0},

then

{bj:

e

F}

is also

infinite. It is also clear that if b

J[N+i,N+i+l)

18 R.M. KAUFFMAN

exists adecreasingtower

A

_nof Borel sets such that

A

N(R)

A

_n and such that

#e

(An)

is finite

n=l

and positiveforeachn. Sinceby hypothesis,

#e(An)

does notapproach 0, thisimplies that

A

isa point mass.

It

follows that thereexists acountable set

{An}

ofpointsofthespectrum of

H

such

that

#e([N,(R))\{An}

0.

Hence,

the same assertion is true for

e"

These points

A

n are

eigenvaluesofH. Since the essential spectrumof

H

is

unbounded,

and themultiplicityof each

A

n isone, it follows that thereexists anunbounded set

{j}

of clusterpointsof

{An}.

(It

should be remarked that the

A

_narenot necessarilyarrangedinincreasing

order.)

Let

n

be the normalizedeigenfunction correspondingtotheeigenvalue

A

n.

Ifc

n

[e,n]

then

e(An)=

ICn

12

But

FAn,e

anon

forsomecomplexnumber

n"

Hence

Ue(e)(An)

[e,FAn,

j

1

-nCn;

thus

n

1/Cn"

Thus

2

IIFAn,ell

2

IInll2/ICnl;

then

#e(An

IIFAn,ellwe(A

n)

IIWnll2.

Let

j

be acluster pointof

{An}.

Let

{Ar}

converge

to

j.

[Ar,As

0forr

#

s.

But

on

any compact interval

[0,M],

if

I([0,M])

is the characteristic function of

[1,M],

then Ascoli’s theorem guarantees that

{R([0,M])Ar

}

has a cluster point

gM

in

L

_2.

But

if 7_r

[gM,I([0,M])Ar],

then the sequence

7r

is square summable and

7r-IIl([0,M])

A

2

r

convergesto 0. Butsince

Ar

is normalized in

L2,

and

rAr ArAr

we seesincethesequence

A

_ris bounded that

IIA

I1(R)

is also bounded. It follows that foreach

j,

there exists atleast one

r

(actually

infinitely

many)

Ar(j

such that

IIWAr(j

)112

<_

1/j

9",

and such that

Ar(j)- jl

<

1. It follows that the sequence

Ar(j)

approaches infinity, and that if A

{At(j)},

#e(A)

is finite but

e(A\[-N,N])

>

0forallN. The theoremisproved.

5.

DIAGONALIZATION

IN

L2(X,p

In

this section we discuss diagonalization of

p(A),

not just

P(A)P(Se)

and show the equivalence ofcompactness of

P(A)

in

B(Y,

L2)

and diagonalization in

B(Y,Z)

discussed in the introduction.

For

situations where the embedding of

Y

into

L

₂is not compact, these properties

generally hold forsomebut not all

A. In

applications to partial differential equations,it is often

true that

P(A)

is known by other means to be compact for all bounded

A;

an example is the

situation of Theorem 5.5. When

A

is unbounded but the embedding from

Y

into

Z

is not compact, compactness and diagonalization become delicate properties of A as we see for an

example by theresultsofsection4.

Remark."The difference between thefollowing definitionand Definition 3.2is onlythat we

diagonalize

P(A)r(H),

not just

P(A)P(Se)r(H

).

For completeness,

however,

the definitionisgiven

EIGENFUNCTION APPROXIMATIONS FOR CONTINUOUS SPECTRUM OPERATORS 19

Definition5.1.

Assume

thehypothesesof Assumptions 1.2and3.1 andEstimate 1.3. Letr

be a boundedcontinuousfunctionfromthe spectrumofHintothe complexes. Let

A

beanyBorel subset of

.

We

say that

Q--P(A)r(H)

is diagonalizable in

B(Y,Z)

with respect to

H

if

B’Qg

E

L2(X,p)for

allgE

Y,

andthe followingistrue:

a)

for every e

>

0thereexists apositive integer k and a finite disjoint family

{Ai}=lof

m subsetsof

A

suchthat thereexists afinite set

{ei}

=1 of orthonormMelementsofthedomainof

k

such that

Ai,

A

andsuch

that,

injecting

Q0

H

N and a finite set ofreal numbers

{Ai,j}i=

1

canonicallyinto

W’, II{Q

ii

=1’=1

#ej(Ai

r

(i,j)Ri,j,ej}(0)ll

Z

<

ellOlly

for all

Y,

where

b)

Ai,

is in the complement of the exceptional set for

ej

of Theorem 1.8, so that in

particular

F%i,j,e

j

isasinTheorem 1.8and

B’FA..,e.

L2(X’P)’

andwhere

,J

c)

RAi,j,ej(_

)

B’

GAi,j,ej(A)GAi,j,e

for any E

,

where

GAi,j,ej

FA’,j"e’/IIB’FA"elI2’j

and where

B’GA,

_e denotes the complex conjugate of the functional

B’GA,e.

Let

M

beafamilyof boundedcontinuousfunctions fromthe spectrumoftherestrictionof Hto S

einto

C. Let

A

bea Borel subset of. Let

QM,A={P(A)r(H):rEM}"

We

say that

QM,A

is

simultaneousl.]

diagonalizablein

B(Y,Z)

withrespect to

H

ifforevery

>

0thereexists

Ai,

Ai,

jand

ej

as

above,

whichare independent of r,suchthat

II{Q-

i=lj

_=I

#ej(Ai)(Ai,j)RAi,j,ej}(O)llZ

<-

ellOlly

for

MI

e Y

and all

Q

e

QM,6"

Remark: The implication that diagonalizability implies compactness in

B(Y,Z)

follows

from thefact that any operatorwith finitedimensional range iscompact, andthat the compact

operators areadosed subsetof

B(Y,Z).

Theorem 5.2.

Suppose

that

e

hjpotheses

of

Asumption

.1

and 1.

hold,

and that

Estimate1.3holds.

Suppose

thatZ contains

L2(X,p),

and thatthe injection

from

L2(X,p

into

Z

is continuous.

Let

Ey

denote

tAe

injection

from

Y

into

L2(X,p),

and

E

Z

denote the injection

from

Lf(X,p)

intoZ. Then

i)

PEy/s

compactin

B(Y,L2)

if

and only

i.f

P

is

dia7onalizable

in

B(Y,Z)

uith respect to

H,

here P

p(A)

or

P(A)P(Se);

frther,

EZPEy

/s compact in

B(Y,Z) if

and only

ifPEy

/s compactin

B(Y,L2);

ii)

Suppose

that

PEy/s

compactin

B(Y,L2).

Then

for

anybounded continuous

.function

r

from

thespectrum

of

H

intothecomplezes, andany Borelsubset

of

,

including

itself,

Pr(H)/s

diagonalizable ith respect to H.

Furthermore,

if

M

is a set

of

continuous

fnctions

from

the spectrum

of

H

into

f,

and

M

is uniformly bounded and

efuicontinuous

on

A,

then

QM,A

/s

20 R.M. KAUFFMAN

Proof:

Weconsiderthe case

P

P(A).

Theother caseisdone thesame way,except that

it is easier. Notethat

L2(X,p

isthedirect sum ofcountably many subspaces S

ei

wheree is in

the domain of H

N,

and

Ileill

1. If

P(A)Ey

is compact in

B(Y,L2)

an elementary argument

showsthatfor everye

>

0thereexistsa positiveintegermsuchthat

[[P(A)Ey-i

=IP(Sei)P(A)Ey[[B(Y,L2)

<

e.

Also,

thereexists a tower

{j}

ofBorel measurablesubsetsof[_{such that}

%i(\U-

I

j)--

O for

all i_<m, and such that, for

AE

{j,

[[B’FA,ei[[

2<_j

for all i_<m. Let

j=

j\j_l.

Let

Aj

ANj.

Again using the fact that

P(A)Ey

is compact, we see that for the above e, there

exists aJsuchthat

[[jj=IP(Ai)Ei

=IP(Sei)EY-i

=IP(Sei)P(A)EyIlB(Y,L2)<

e.

Hence,

since

r(H)

isuniformlyboundedbysome constant

I’

inoperatornorm for r E

M,

it

follows that for allr E

M,

[IP(A)r(H){Ey-

jj=IP(Aj)Ei

=IP(Sei)EY}[[B(Y,L2)

<

2el’.

The implication that compactness implies diagonalizability follows immediately, upon using the implication

ii)iii)

ofTheorem 3.3foreachS

ei

for

<_

mtogether withthefact that the injection

from

L

₂into

Z

isbounded.

Suppose

that

P(A)Ey

is notcompact. Then thereexists aboundedsequence

Yn

in

Y

such that

P(A)yn

has noconvergent subsequencein

L

_2. Thereexistsaweakly

convergent

subsequence

to the sequence

P(A)Yn,

which we assume without loss ofgenerality is the original sequence; suppose

P(A)y

_nconverges weakly tog. If

P(A)

isdiagonalizablein

B(Y,Z),

thenbythe remark

preceding the theorem

EZP(A)Ey

is compact in

B(Y,Z);

thus

EZP(A)yn

has a subsequence,

whichagain we assumeistheoriginal sequence,whichconverges to

EZg

in

Z

andthereforeinY’.

EZP(A)yn(Yn)

convergesto

[[g[[,"

byanelementary argument.

But

Now

EZP(A)yn(Yn) --[[P(A)Yn[[

,-

whichthen

converges

to

[[g[[2.-

By

anotherelementary argument,it

follows that

P(A)y

_n

converges

to gin

L2,

acontradiction.

Hence

EZP(A)Ey

isnot compact, so

that

p(A)

is notdiagontizablein

B(Y,Z).

Thefirstassertion isproved; becausewehaveproved that diagonalizability implies compactness in

B(Y,Z)

which implies compactness in

B(Y,L2)

which impliesdiagonalizability.

Remark:

We

nowtesttheresults againstthe abstract SchrOdingerequation, whichis more difficult

than,

for example, theabstract heat equation,because dampingisnotpresent.

Corollary5.3.

Suppose

that the h./potheses

of

Theorem5.hold.

Suppose

that

P(A)Ey/s

a compact operatorin

B(Y,Z).

Then

for ever

pair

T,e

of

positive real

numbers,

there ezists a

finite

set

{F,i}

of

elements

of

Z

such that

H’Fi ’iF’i

for

some

"i

Afl spectrum

(H),

and such that

IIP(A)eiHt-

l

e-i’t

EIGENFUNCTION APPROXIMATIONS FOR CONTINUOUS SPECTRUM OPERATORS 21

Proof."

By

Theorem 5.2,

{P(AI3[-n,n])e

iHt"

Itl

<_

T)

is simultaneously diagonalizablein

B(Y,Z),

since

{e

iAt" A

E

[-n,n],

Itl

<_

T}

isequicontinuous. Using the compactnessof

EZP(A)Ey

again as inthe proof ofTheorem 5.2, we see that for any e

>

0 thereexists a positive integern

suchthat

IIP(Af[-n,n])e

iHt

P(A)eiHtlIB(y,z)

<

e. The resultfollowsfrom Theorem 5.2.

Ezample:

In

Example 2.3,let

Y

{0

E

L2(n)l

(ll=+l)V(i=

/x

1)M0

L2(n),

and

II011y

-II(Ixl

9+1)’r(i_1

02/0x-

1)M0119..

This example yidds the nextcorollary.

Remark Inthe next result,welookfor situationswhere the approximation property holds for the whole operator e

iHt,

so that wemay take A Ill. Wefind that this is possiblein great generality, but at the price ofsubstantially restricting the space

Y

and increasing the space

Z,

which has the practical effect of substantially weakening the error estimate

from,

say, the

situationofSection 4.

Corollary 5.4. Suppose that 7 is a partial

differential

ezpression on

n

such that the

coefficients

of

7"areinfinitely

differentiable,

and such that anyderivative

of

a

coefficient

hasat most

polynomial growth at infinity.

Suppose H

is any self-adjoint operator in

L2(

n)

such that the rapidly decreasing

functions

are contained inthedomain

of

H,

andsuch that

H