Properties of the Discrete and Continuous Spectrum of Differential Operators

Spe trum of Dierential Operators

ANDREAS ENBLOM

Do toral Thesis

TRITA-MAT-09-MA-02

ISSN1401-2278

ISRNKTH/MAT/DA09/01-SE

ISBN978-91-7415-227-2

KunglTekniskahögskolan

InstitutionenförMatematik

10044Sto kholm

SWEDEN

AkademiskavhandlingsommedtillståndavKunglTekniskahögskolanframlägges

till oentliggranskning för avläggandeavteknologie doktorsexamen i matematik

fredagen den 27 februari 2009 klo kan14.00 i salF3, KunglTekniskahögskolan,

Lindstedtsvägen26,Sto kholm.

©AndreasEnblom,2009

Thisthesis ontainsthrees ienti papersdevotedto thestudyofdierentspe tral

theoreti alaspe tsofdierentialoperatorsinHilbertspa es.

Therstpaper on erns themagneti S hrödingeroperator(i∇ + A)^{2 in}L²(Rⁿ)^{. It}

isprovedthat given ertain onditions onthe de ayof A^{,the set} [0, ∞)^{is anessential}

supportof theabsolutely ontinuouspart of the spe tral measure orrespondingto the

operator.

These ondpaper onsidersaregulard-dimensionalmetri treeΓ^{anddenesS hrödin-}

geroperators −∆ − V ^{onit. Here,}V ^{is asymmetri ,}non-negativepotentialonΓ^{. Itis}

assumedthatV ^{de ayslike}|x|^{−γ atinnity,where}1 < γ ≤ d ≤ 2, γ 6= 2^{. Aweak oupling}

onstantα^{isintrodu edinfrontof}V^,andtheasymptoti softhebottomofthespe trum asα→ 0+^{isdes ribed.}

Thethird, and last, paperrevolvesaroundfourth-order dierentialoperators inthe

spa eL²(Rⁿ)^,wheren= 1 ^orn= 3^{. In}parti ular,theoperator(−∆)²− C|x|⁻⁴− V (x)

isstudied,whereC^{isthesharp onstantinthe}Hardy-Relli hinequality. ALieb-Thirring inequalityfor thisoperator isproved,and asa onsequen e aSobolev-typeinequalityis

obtained.

Dennaavhandlinginnehållertrevetenskapligaartiklarsomstuderarolikaspektralte-

oretiskaegenskaperfördierentialoperatoreriHilbertrum.

Den förstaartikelnberör denmagnetiskaS hrödingeroperatorn(i∇ + A)^{2 i}L²(Rⁿ)^.

GivetvissavillkorpåavtagandethosA^{,bevisasdetattmängden}[0, ∞)^{ärettväsentligt}

stödfördenabsolutkontinuerligadelenhosdetspektralmåttsomhörtilloperatorn.

Denandraartikelnbetraktarettreguljärtd-dimensionelltmetrisktträdΓ^{o hdenie-}

rarS hrödingeroperatorn−∆ − V ^{pådet.Härär}V ^ensymmetrisk,i ke-negativpotential påΓ^.DetantasattV ^avtarsom|x|^{−γ vid}oändligheten,där1 < γ ≤ d ≤ 2, γ 6= 2^{,o hen}

svagkopplingskonstantα^{införsframför}V^{.Sedanbeskrivs}asymptotikenförunderkanten avspektretdåα→ 0+^.

Dentredje,o hsista,artikelnbehandlarfjärdeordningensdierentialoperatorerirum-

metL²(Rⁿ)^,därn= 1^ellern= 3^{.Isynnerhetstuderasoperatorn}(−∆)²−C|x|⁻⁴−V (x)^,

därC^{ärdenskarpakonstanteni}Hardy-Relli holikheten.EnLieb-Thirringolikhetförope- ratornbevisas,o hfråndennaföljer enSobolevolikhet.

Contents v

Prefa e vii

Introdu tion and Summary 1

1 Ba kground 3

1.1 HilbertSpa es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Self-AdjointOperators . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Generi ExampleI:Multipli ationOperators . . . . . . . . . . . . . 10

1.4 Generi ExampleII:TheDiri hletLapla ian . . . . . . . . . . . . . 12

2 Dis ussionofPaperI 15 2.1 Magneti operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 De ompositionofSpe tralMeasures. . . . . . . . . . . . . . . . . . 17

2.3 EssentialSupportsof Spe tralMeasures . . . . . . . . . . . . . . . 18

3 Dis ussionofPaperII 21 3.1 RegularMetri Trees . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 WeakCouplingAsymptoti s . . . . . . . . . . . . . . . . . . . . . . 23

4 Dis ussionofPaperIII 25 4.1 Tra eofSelf-AdjointOperators . . . . . . . . . . . . . . . . . . . . 25

4.2 Fourth-OrderOperators . . . . . . . . . . . . . . . . . . . . . . . . 25

4.3 Criti alExponents . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Bibliography 29 I On the Absolutely Continuous Spe trum of Magneti S hrödingerOperators 31 1 Introdu tionandMainResult . . . . . . . . . . . . . . . . . . . . . 33

2 TheAbsolutelyContinuousSpe trum . . . . . . . . . . . . . . . . . 34

4 Appli ationtotheMagneti Hamiltonian. . . . . . . . . . . . . . . 54

Referen es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

II S hrödinger Operators on Regular Metri Trees with Long Range Potentials: Weak Coupling Behavior 59 1 Introdu tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2 MainResult . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3 Spe tralTheoryonRegularMetri Trees . . . . . . . . . . . . . . . 64

4 Redu tionto anOperatorontheHalf-Line . . . . . . . . . . . . . . 65

5 EstimatesoftheBottomoftheSpe trum. . . . . . . . . . . . . . . 67

A Appendix: TheFourier-BesselTransform . . . . . . . . . . . . . . . 72

B Appendix: IntegralKerneloftheBirman-S hwingerOperator . . . 77

Referen es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

III Lieb-Thirring Inequalities for Fourth-Order Operators inLow Dimensions 83 1 Introdu tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

2 MainResults. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

3 AnAuxiliaryOperatoronaFiniteInterval . . . . . . . . . . . . . . 89

4 Well-BehavedPotentials . . . . . . . . . . . . . . . . . . . . . . . . 96

5 ProofoftheMainResults . . . . . . . . . . . . . . . . . . . . . . . 104

Referen es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

ThisisthethesisthatsummarizesmyworkforaPhDdegreeinMathemati satthe

Departmentof Mathemati sof theRoyalInstitute ofTe hnology(KTH) between

theyears2003and2008.

The thesis onsists of three s ienti papers, as well as an Introdu tion and

Summary. The purpose of the Introdu tion and Summary is twofold. First, it

des ribestheba kgroundandthemainresultsofthethreepapers.These ond,and

maybeevenmoreimportant,purposeis toprovideanyreaderwhoisnotfamiliar

with thesubje t of spe tral theoryof unbounded operatorsin Hilbert spa es the

ne essaryba kground to understand the three papers. I do not laim that this

introdu tion is omplete, and many of the proofs havebeen left out, but it has

beenmyintentionthatitshould ontaineverythingneeded,andnotmu hmore.

Therst four yearsof my PhDstudies werefunded by the GöranGustafsson

foundation. I amgratefultothefoundationand ommendthemontheirnework

ofsupportingtheSwedishresear h ommunity.

IwouldliketothankmysupervisorAriLaptevformanyusefuldis ussions,for

alwayspushing onwith this thesis, andfor without any hesitationtaking are of

pra ti alandnan ialarrangementsforme.

Oleg Safronov introdu ed me to this subje t and taught me the theoreti al

frameworkuponwhi h mu hofthis thesisisbased. Iamgratefulto himforthat.

Throughouttheyears,I havehadmanymathemati aldis ussionswithRupert

Frank,whohasalsoreadmostofmymanus ripts,andIwishtothankhimaswell.

TheDepartmentofMathemati s at KTHisawonderfuland reativepla eto

workat. IamveryhappytohavesomeofmybestfriendsthereandIamthankful

toallofthem,andespe iallytomyroommateFredrikNordström,formakingthese

pastyearsmu hmorethan`allworkandnoplay'.

Finally,I am in debtto my best friend, and o-author,Tomas Ekholm forhis

onstanten ouragementand enthusiasm.

AndreasEnblom

De ember,2008

This hapter givesabriefsummaryofsomeessentialaspe tsofthetheoryofself-

adjoint operators in Hilbert spa es. For a omplete a ount of this theory, see

[BS87℄. Otherstandardreferen esonthis subje tare[RS80℄and[Dav95℄.

Throughout thisintrodu tion, thereader isassumed to be familiarwith basi

Hilbertspa etheoryaswellasmeasureandintegrationtheory.

1.1 Hilbert Spa es

Let

H

bea omplex,separableandinnite-dimensionalHilbert spa e. Denoteby

(·, ·)

^and

k · k

^{the inner produ t and norm,} respe tively, in

H

. Theexamples of Hilbert spa es are numerous, but in this thesis two of those examplesstand out:

theLebesguespa es

L

²

(X, µ)

^{andtheSobolevspa es}

H

^m

(Ω)

^.

Givenameasurespa e

(X, µ) = (X, A , µ)

^{,thespa e}

L

²

(X, µ)

^{isdenedasthe}

setofmeasurable, omplex-valuedfun tions

f

^on

X

^{su hthat}

kfk

²

:=

Z

X

|f|

²

dµ < ∞.

Twosu h fun tions

f

1 ^and

f

2 ^{are identiedin}

L

²

(X, µ)

^{ifthey oin ide}

µ

^-almost

everywhere. The typi alexample of su h aLebesgue spa e o urs when

X

^{is an}

open subset of

R

ⁿ and

µ

^{is aBorel measure on}

X

^{. Typi ally}

µ

^{is the Lebesgue}

measure

dx

^{,andin this asewewrite}

L

²

(X) = L

²

(X, dx)

^.

It anbeadded that noteverymeasure spa e

(X, µ)

^{is su h that}

L

²

(X, µ)

^is

separableandinnite-dimensional. Butspa essu has

L

²

(Ω)

^{, where}

Ω

^isanopen

subsetof

R

ⁿ,or

L

²

((a, b), φ(x) dx)

^,where

−∞ ≤ a < b ≤ ∞

^and

φ

^{isa ontinuous}

fun tionon

R

,havethisproperty.

Let

Ω

^{be anopen subset of}

R

ⁿ, and

m ≥ 1

^{an integer. We dene the spa e}

H

^m

(Ω) = W

^m,2

(Ω)

^{asthe spa eoffun tions}

u ∈ L

²

(Ω)

^{su h that}

D

^α

u ∈ L

²

(Ω)

foranymulti-index

α ∈ Z

ⁿ+ ^with

|α| ≤ m

^{. Here,the}dierentialexpression

D

^α

u = ∂

^α1

∂x

^α₁¹

∂

^α2

∂x

^α₂²

· · · ∂

^αn

∂x

^αnⁿ

u(x

1

, x

2

, . . . , x

n

)

isinterpretedin thesenseofdistributions,and

|α| = α

¹

+ α

2

+ · · · + α

ⁿ

.

Thenormin

H

^m

(Ω)

^isdenoted

k · k

m,2 ^andisgivenby

kuk

²m,2

= X

1≤|α|≤m

Z

Ω

|D

^α

u|

²

dx + Z

Ω

|u|

²

dx.

Thetheory ofSobolevspa esisthoroughly dis ussedin[AF03℄. Furthermore,the

norm

k · k

m,2 ^{is equivalent to the following norm, where only the highest order}

dierentialtermsarein ludedinthesum:

u 7→ X

|α|=m

Z

Ω

|D

^α

u|

²

dx + Z

Ω

|u|

²

dx.

Theset of ompa tly supportedsmoothfun tionson

Ω

^{isdenoted by}

C

₀^∞

(Ω)

^,

andwehavethefollowing hainofin lusions:

C

₀^∞

(Ω) ⊂ H

^m

(Ω) ⊂ L

²

(Ω) .

Also,

C

₀^∞

(Ω)

^{is densein}

L

²

(Ω)

^{, whi h makes}

H

^m

(Ω)

^{dense in}

L

²

(Ω)

^{. However,}

in general

C

₀^∞

(Ω)

^{is not densein}

H

^m

(Ω)

^{, with respe t to thenorm}

k · k

m,2^{. We}

thereforeintrodu ethespa e

H

₀^m

(Ω)

^{asthe losurein}

H

^m

(Ω)

^of

C

₀^∞

(Ω)

^{. Thatsaid,}

when

Ω = R

^{n, fun tions from}

H

^m

(Ω)

^{an a tuallybe}approximatedbyfun tions in

C

₀^∞

(Ω)

^{inthesensethat}

H

₀^m

(R

ⁿ

) = H

^m

(R

ⁿ

) .

Operators

Anoperator

H

^isalinearmap

D (H) → H

^,where

D (H)

^{isalinear,densesubset}

of

H

. The set

D (H)

^{is alled the domain of}

H

^{. Theoperator}

H

^{is said to be}

bounded ifthere isa onstant

C > 0

^{su hthat}

kHfk ≤ Ckfk, f ∈ D (H) .

Note that by the above denition, general operators are neither assumed to

bebounded noreverywheredened. Thereasonfornotrequiringoperators tobe

everywheredened willbegivenbythe losed-graphtheorem,below.

Of parti ular interest are the losed operators. Given an operator

H

^{in the}

spa e

H

,dene thegraphof

H

^tobetheset

Γ(H)

^givenby

Γ(H) = {hf, Hfi ; f ∈ D (H)} .

Clearly,

Γ(H)

^{isalinearsubsetof}

H

². Wesaythattheoperator

H

^{is losed ifthe}

graph

Γ(H)

^{isa losedsubsetof}

H

².

Theorem 1.1.1 (The losed-graph theorem). Let

H

^{be an operator in}

H

with domain

D (H)

^{. Supposethattwoof the following onditionsaresatised:}

(i) Theoperator

H

^isbounded.

(ii) Theoperator

H

^{is losed.}

(iii) Theoperator

H

^{iseverywhere dened, i.e.}

D (H) = H

^.

Thenthe third onditionisalso satised.

Sin eadenselydened,boundedoperator analwaysbeextendedby ontinuity

totheentirespa e

H

,thereisnolossofgeneralityindemandingthatallbounded operators are everywhere dened. In parti ular, bounded operators are always

losed.

Supposethattheoperator

H

^{isnot losed. Let}

Γ(H)

^{bethe losureof}

Γ(H)

ⁱⁿ

H

². Wheneverthereisanoperator

H

ⁱⁿ

H

su hthat

Γ(H)

^isthegraphof

H

^,we

saythat

H

^{is losable andthat}

H

^{isthe losure of}

H

^.

Note that not all operators are losable, but if

H

^{is losable, then}

H

^{is an}

extensionof

H

^,meaningthat

D H

⊃ D (H)

^and

Hf = Hf, f ∈ D (H) .

Infa titturnsoutthat

H

^{isa tuallytheminimal losedextensionontheoperator}

H

^{,i.e. anyother losedextension of}

H

^{isanextensionof}

H

^.

Operators ofspe ialinterestaretheorthogonal proje tions in

H

^{. Let}

P

^bean

orthogonalproje tionand

H

₁itsrange. Re allthat

H

₁isa losedsubspa eof

H

, andthattheproje tionisgivenby

P f =

 f, f ∈ H

1

, 0, f ⊥ H

1

.

Thedimensionof

H

₁ is alledtherank of

P

^.

Spe trum

Let

ρ (H)

^{be the set of}

z ∈ C

^{su h that the operator}

H − z

^{is invertible and}

theinverse

(H − z)

^{−1 is a bounded operator withdomain}

H

. The omplement,

σ (H) := C \ ρ (H)

^{,is alledthespe trum of}

H

^.

The set

ρ (H)

^{is an open subset of}

C

, and the bounded operator

(H − z)

⁻¹

denedfor

z ∈ ρ (H)

^{is alled theresolvent of}

H

^atthepoint

z

^{. Inparti ularthe}

spe trum

σ (H)

^{is a losed subset of}

C

. Note that for any

z ∈ ρ (H)

^{, there is a}

onstant

c > 0

^{su hthat}

k(H − z)fk ≥ ckfk, f ∈ D (H) .

^(1.1)

Apoint

z ∈ C

^{is alledaneigenvalueoftheoperator}

H

^ifthereisa

f ∈ D (H)

su h that

Hf = zf

^{. Theve tor}

f

^{is saidtobean}eigenve tor orresponding to

z

^.

Clearly, everyeigenvalue of

H

^{belongs to}

σ (H)

^{. Thedimension of the subspa e}

spanned by the eigenve tors orresponding to

z

^{is alled the} multipli ity of the eigenvalue

z

^.

Thesubset

σ

d

(H)

^of

σ (H)

^{onsistingofisolated}eigenvaluesofnitemultipli ity is alledthedis retespe trum. Wealsodenetheessentialspe trum

σ

e

(H)

^asthe

non-dis retepartofthespe trum. Hen e

σ

e

(H) = σ (H) \ σ

d

(H)

^.

Theabovedenitionofspe trumisvalidforanydensely-denedoperator. But

it turns out that from a spe tral-theoreti al point of view, the only interesting

operatorsarethe losed operators:

Proposition 1.1.2. Supposethat

H

^{is an operator in}

H

. Then

σ (H) 6= C

^only

if

H

^{is losed.}

Finally,pleasenotethatthereisabigdieren ebetweenthe aseofageneral,

innite-dimensional Hilbert spa e and the orresponding nite-dimensional ase.

Inthe nite-dimensional ase, all operators are bounded, and have only dis rete

spe trum.

Unitary Equivalen e

ConsidertwoHilbertspa es

G

and

H

. Alinear,bije tivemap

U : G → H

^issaid

tobeaunitaryoperator,orunitaryequivalen e,if

(f, g) = (U f, U g), f, g ∈ G .

Itis learthatfromanabstra tpointofview,su haunitaryequivalen eprovides

anidenti ationoftheHilbert spa es

G

and

H

.

Suppose that

G

^{isanoperator in}

G

and that

H

^{isanoperator in}

H

. Ifthere isaunitaryequivalen e

U : G → H

^{su h that}

U D (G) = D (H)

^and

U Gf = HU f, f ∈ D (G) ,

thenwesaythattheoperators

G

^and

H

^{areunitarily equivalentandwewritewrite}

G = U

⁻¹

HU

^{. Inthat ase,theoperators anbeviewedasoneandthesame,and}

allthespe tralpropertiesof thetwooperators oin ide.

1.2 Self-Adjoint Operators

Theoperators ofthemostpra ti alimportan eare theself-adjoint operators. In

nite-dimensionalve torspa es,they orrespond totheHermitian, orsymmetri ,

matri es. Like in the nite-dimensional ase there is a spe tral theorem for the

Symmetry

Anoperator

H

^{issaidtobesymmetri if,forany}

f, g ∈ D (H)

^,

(Hf, g) = (f, Hg).

Notethatinthe aseofanite-dimensionalHilbertspa e

H

,thisisthedenition ofaHermitianmatrix. However,in theinnite-dimensional ase,where there are

unbounded operators,symmetryisamu h weaker onditionthanself-adjointness,

aswewillsee. Forboundedoperators,thenotionsofsymmetryandself-adjointness

oin ide.

An importantpropertyofsymmetri operators isthat theyalwaysgiveriseto

losedsymmetri operators.

Proposition1.2.1. Anysymmetri operator

H

^{is losable. The losure}

H

^isalso

symmetri .

Self-Adjointness

If

H

^{is an operator, theset of all}

g ∈ H

^{for whi h there is a}

h ∈ H

^{su h that}

(Hf, g) = (f, h)

^{for any}

f ∈ D (H)

^{is denoted by}

D (H

^∗

)

^{. Sin e}

D (H)

^{is dense}

in

H

, the element

h

^{is uniqueand is denoted by}

H

^∗

g

^{. If}

D (H) = D (H

^∗

)

^and

Hf = H

^∗

f

^forany

f ∈ D (H)

^,theoperator

H

^issaidtobeself-adjoint.

It turns out that self-adjoint operators are always symmetri and have real

spe trum. Infa t,thisisa omplete hara terizationofself-adjointness:

Proposition 1.2.2. Let

H

^{be an operator in}

H

. Then

H

^is self-adjoint if and onlyif

H

^{issymmetri andsatises}

σ (H) ⊂ R.

Note in parti ular that itthis together with Proposition 1.1.2 showsthat any

self-adjointoperatoris losed.

It is now timeto statethespe traltheorem for self-adjoint operators. Letus

startbyintrodu ingthenotionofaspe tral measure.

Denition 1.2.3. Let

B

denote the Borel subsets of the real line. A spe tral measure isafun tion

E

^from

B

intothesetoforthogonalproje tionson

H

su h that:

(i)

E(R) = I

^.

(ii) The fun tion

E

f

: B ∋ A 7→ (E(A)f, f) ∈ R

^{is a (nite) measure for any}

f ∈ H

^.