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• DISSERTATION

Titel der Dissertation

Oscillation Theorems for Semi-Infinite and Infinite

Jacobi Operators

Verfasserin

Kerstin Ammann

Doktorin der Naturwissenschaften (Dr. rer. nat.)

Wien, im Jänner 2013

Studienkennzahl lt. Studienblatt: A 091 405 Dissertationsgebiet lt. Studienblatt: Mathematik Betreuer: Univ.-Prof. Dipl.-Ing. Dr. Gerald Teschl

• Abstract

The Jacobi difference equation (JDE) plays an important role (not just) in math-

ematical physics: e.g., it containes the one-dimensional discrete Schrödinger

equation as a special case and is intimately related to the theory of orthogonal

polynomials as well as to continued fractions.

While classical oscillation theory for Jacobi operators puts the sign-changes of

solutions of one single operator at the centre of consideration, we compare the

number of sign-changes of solutions of two different Jacobi operators. We show

that this difference equals the number of weighted sign-changes of the Wronskian

of those solutions.

The key discovery in oscillation theory, which goes back to the work of Sturm,

is the fact that for any real z the number of sign-changes of a solution u(z)

equals the number of eigenvalues of the operator below z. Our theorem refines

this observation for the JDE by showing that the number of weighted sign-

changes of the Wronskian equals the difference of the number of eigenvalues

of the operators in the corresponding interval. The main advantage of this

approach is that our theorem is also applicable in gaps of the essential spectrum

above its infimum, where the classical theorem breaks down (since the solutions

are oscillatory, but the Wronskian isn’t).

This theorem is proven for compact, sign-definite perturbations of the potential

of Jacobi operators on the line and on the half-line. For the finite case, we extend

earlier work for perturbations of the potential to perturbations of all coefficients.

Moreover, we show that this idea carries over to the leading principal minors of

Jacobi matrices, which exhibit the same sign pattern as a solution at 0.

• Zusammenfassung

Die Jacobi Differenzengleichung (JDG) spielt (nicht nur) in der mathemati-

schen Physik eine wichtige Rolle: z.B. beinhaltet sie die eindimensionale diskrete

Schrödingergleichung als Spezialfall und ist eng mit der Theorie der orthogona-

len Polynome sowie den Kettenbrüchen verknüpft.

Während die klassische Oszillationstheorie für Jacobi Operatoren die Vorzei-

chenwechsel der Lösungen eines einzigen Operators ins Zentrum ihrer Betrach-

tungen stellt, vergleichen wir die Anzahl der Vorzeichenwechsel von Lösungen

zweier verschiedener Jacobi Operatoren. Wir zeigen, dass diese Differenz der An-

zahl der gewichteten Vorzeichenwechsel der Wronski Determinante der beiden

Lösungen entspricht.

Die zentrale Entdeckung der Oszillationstheorie geht zurück auf Sturm und be-

sagt, dass für jedes reelle z die Anzahl der Vorzeichenwechsel einer Lösung u(z)

der Anzahl der Eigenwerte des Operators unterhalb von z entspricht. Unser

Theorem entwickelt diese Beobachtung für die JDG dahingehend weiter, dass

es zeigt, dass die Anzahl der gewichteten Vorzeichenwechsel der Wronski De-

terminante der Differenz der Anzahl der Eigenwerte der beiden Operatoren im

zugehörigen Intervall entspricht. Der Vorteil dabei ist, dass unser Theorem auch

in Lücken des wesentlichen Spektrums überhalb seines Infimums anwendbar ist,

im Gegensatz zum klassischen Theorem (da die Lösungen hier oszillatorisch

sind, die Wronski Determinante aber nicht).

Dieses Theorem wird für kompakte, vorzeichenbestimmte Störungen des Poten-

tials von singulären Jacobi Operatoren, wie auch von Jacobi Operatoren mit ei-

nem regulären Endpunkt, bewiesen. Für den endlichen Fall erweitern wir frühere

Arbeiten über Störungen des Potentials auf Störungen aller Koeffizienten. Wei-

ters zeigen wir, dass sich diese Idee auch auf die führenden Hauptminoren von

Jacobi Matrizen übertragen lässt, da sie das selbe Vorzeichenmuster aufweisen

wie eine Lösung bei 0.

• Acknowledgments

First of all I want to thank Gerald Teschl, who already supervised my diploma

thesis and from whom I’ve learnt a lot in the past years. In particular I thank

him for his support and for all the helpful suggestions he made. Moreover, I

thank Günther Hörmann for his support and that he motivated me to solve

(1.20) and (1.21), and Helge Krüger (Caltech) that he discussed the continu-

ous case with me. Further, I thank the referees Fritz Gesztesy (University of

Missouri-Columbia) and Jussi Behrndt (Graz University of Technology) for the

time they devoted to my thesis and the valuable suggestions they made.

This thesis was funded by the Austrian Science Fund (FWF) as part of Ger-

ald’s project Spectral Analysis and Applications to Soliton Equations. This al-

lowed me moreover to present my work at international conferences, what led to

new research projects with Uzy Smilansky (Weizmann Institute of Science and

Cardiff University) and Ram Band (University of Bristol) since they evoked my

interest for nodal domains on graphs.

I want to thank my friends, whose support was important for me, in particular

also those at the math department and my colleagues Alice Mikikits-Leitner

(TU Munich) and Jonathan Eckhardt; and I thank Jürgen Pfeffer (Carnegie

Mellon University) for his ongoing encouragement and aid.

• Contents

1 Introduction 1

2 Preliminaries 13

2.1 Linear operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Spectra and resolvents . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3 Operator convergence . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4 Green function and Weyl solutions . . . . . . . . . . . . . . . . . 19

2.5 Weyl m-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3 Weighted nodes 27

3.1 Wronskian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2 Discrete Prüfer transformation . . . . . . . . . . . . . . . . . . . 31

3.3 Difference of the Prüfer angles . . . . . . . . . . . . . . . . . . . 33

4 Finite Jacobi matrices 41

5 Determinants 47

5.1 Solutions and leading principal minors . . . . . . . . . . . . . . . 48

5.2 A Wronskian of determinants . . . . . . . . . . . . . . . . . . . . 49

5.3 Proof of Sturm’s theorem by Jacobi’s theorem . . . . . . . . . . . 52

5.4 A short note on Sylvester’s criterion . . . . . . . . . . . . . . . . 55

6 Triangle inequality and comparison theorem 59

7 Criteria for the oscillation of the Wronskian 63

8 Approximation 71

8.1 . . . of infinite matrices and their spectra . . . . . . . . . . . . . . 71

8.1.1 Open and half-open spectral intervals . . . . . . . . . . . 74

8.1.2 A point . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

8.2 . . . of the Wronskian with suitable boundary conditions . . . . . . 81

8.2.1 Nodes on the half-line . . . . . . . . . . . . . . . . . . . . 81

8.2.2 Nodes on the line . . . . . . . . . . . . . . . . . . . . . . . 83

8.3 . . . of the Wronskian with foreign boundary conditions . . . . . . 84

• 9 Below the essential spectra 91

9.1 The semi-infinite case . . . . . . . . . . . . . . . . . . . . . . . . 92

9.2 The infinite case . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

10 Semi-infinite Jacobi operators 99

10.1 A first theorem on the half-line . . . . . . . . . . . . . . . . . . . 100

10.2 Main theorem for semi-infinite operators . . . . . . . . . . . . . . 104

10.3 A proof for finite-rank perturbations . . . . . . . . . . . . . . . . 111

11 Infinite Jacobi operators 113

11.1 A first theorem on the line . . . . . . . . . . . . . . . . . . . . . . 113

11.2 The finite-rank case . . . . . . . . . . . . . . . . . . . . . . . . . 117

11.3 Main theorem for infinite operators . . . . . . . . . . . . . . . . . 119

A Linear interpolation 125

A.1 Derivative of the Prüfer angle . . . . . . . . . . . . . . . . . . . . 128

Bibliography 132

• Chapter 1

Introduction

In this thesis we present new oscillation theorems for a particular discrete equa-

tion, namely the Jacobi difference equation (JDE),

τu = zu, (1.1)

where z ∈ R,

τ : `(Z)→ `(Z),

u(n) 7→ (τu)(n) = a(n)u(n+ 1) + a(n− 1)u(n− 1) + b(n)u(n) (1.2)

= ∂(a(n− 1)∂u(n− 1)) + (b(n) + a(n) + a(n− 1))u(n),

and where `(I) = {ϕ | ϕ : I ⊆ Z→ R} is the space of real-valued sequences and ∂ϕ(n) = ϕ(n+ 1)− ϕ(n) is the usual forward difference operator.

The JDE can be viewed as the discrete counterpart of the famous Sturm–

Liouville differential equation, τu = zu, where

τ = 1

r(x)

( − d

dx p(x)

d

dx + q(x)

) . (1.3)

Setting a = 1 (that is p = r = 1 in the continuous case) we obtain the one-

dimensional Schrödinge

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