Spectral Theory in Hilbert Space

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4 0. INTRODUCTION • Can the equation be reconstructed if the spectrum is known? If not, what else must one know? If different equations can have the same spectrum, how many different equations? What do they have in common? Questions like these are part of what is called inverse spectral theory. Really satisfactory answers have only been obtained for the equation −u ′′ + qu = λu, notably by Gelfand and Levitan in the early 1950:s. Pioneering work was done by Göran Borg in the 1940:s. • Another aspect of the first point is the following: Given a ‘base’ equation (corresponding to a ‘free particle’ in quantum mechanics) and another equation, which outside some bounded region is close to the base equation (an ‘obstacle’ has been introduced), how can one relate the eigenfunctions for the two equations? The main questions of so called scattering theory are of this type. • Related to the previous point is the problem of inverse scattering. Here one is given scattering data, i.e., the answer to the question in the previous point, and the question is whether the equation is determined by scattering data, whether there is a method for reconstructing the equation from the scattering data, and similar questions. Many questions of this kind are of great importance to applications.
CHAPTER 1 Linear spaces This chapter is intended to be a quick review of the basic facts about linear spaces. In the definition below the set K can be any field, although usually only the fields R of real numbers and C of complex numbers are of interest. Definition 1.1. A linear space or vector space over K is a set L provided with an addition +, which to every pair of elements u, v ∈ L associates an element u + v ∈ L, and a multiplication, which to every λ ∈ K and u ∈ L associates an element λu ∈ L. The following rules for calculation hold: (1) (u + v) + w = u + (v + w) for all u, v and w in L.(associativity) (2) There is an element 0 ∈ L such that u + 0 = 0 + u = u for every u ∈ L. (existence of neutral element) (3) For every u ∈ L there exists v ∈ L such that u+v = v +u = 0. One denotes v by −u. (existence of additive inverse) (4) u + v = v + u for all u, v ∈ L. (commutativity) (5) λ(u + v) = λu + λv for all λ ∈ K and all u, v ∈ L. (6) (λ + µ)u = λu + µu for all λ, µ ∈ K and all u ∈ L. (7) λ(µu) = (λµ)u for all λ, µ ∈ K and all u ∈ L. (8) 1u = u for all u ∈ L. If K = R we have a real linear space, if K = C a complex linear space. Axioms 1–3 above say that L is a group under addition, axiom 4 that the group is abelian (or commutative). Axioms 5 and 6 are distributive laws and axiom 7 an associative law related to the multiplication by scalars, whereas axiom 8 gives a kind of normalization for the multiplication by scalars. Note that by restricting oneself to multiplying only by real numbers, any complex space may also be viewed as a real linear space. Conversely, every real linear space can be ‘extended’ to a complex linear space (Exercise 1.1). We will therefore only consider complex linear spaces in the sequel. Let M be an arbitrary set and let C M be the set of complex-valued functions defined on M. Then C M , provided with the obvious definitions of the linear operations, is a complex linear space (Exercise 1.2). In the case when M = {1, 2, . . . , n} one writes C n instead of C {1,2,...,n} . An element u ∈ C n is of course given by the values u(1), u(2), . . . , u(n) 5
Page 1 and 2: Spectral Theory in Hilbert Space Le
Page 3: Preface The aim of these notes is t
Page 6 and 7: iv CONTENTS Exercises for Chapter 1
Page 8 and 9: 2 0. INTRODUCTION cos( √ λ t), i
Page 12 and 13: 6 1. LINEAR SPACES of u so one may
Page 14 and 15: 8 1. LINEAR SPACES Exercises for Ch
Page 16 and 17: 10 2. SPACES WITH SCALAR PRODUCT on
Page 18 and 19: 12 2. SPACES WITH SCALAR PRODUCT Pr
Page 20 and 21: 14 2. SPACES WITH SCALAR PRODUCT Ex
Page 22 and 23: 16 3. HILBERT SPACE Given a normed
Page 24 and 25: 18 3. HILBERT SPACE Two subspaces M
Page 26 and 27: 20 3. HILBERT SPACE ˆvn = limj→
Page 29 and 30: CHAPTER 4 Operators A bounded linea
Page 31 and 32: 4. OPERATORS 25 Proposition 4.2. A
Page 33 and 34: 4. OPERATORS 27 In the rest of this
Page 35 and 36: 4. OPERATORS 29 must both be 0. Hen
Page 37 and 38: CHAPTER 5 Resolvents We now conside
Page 39: EXERCISES FOR CHAPTER 5 33 Analytic
Page 42 and 43: 36 6. NEVANLINNA FUNCTIONS However,
Page 44 and 45: 38 6. NEVANLINNA FUNCTIONS Proof. B
Page 46 and 47: 40 7. THE SPECTRAL THEOREM function
Page 48 and 49: 42 7. THE SPECTRAL THEOREM as The r
Page 50 and 51: 44 7. THE SPECTRAL THEOREM as a clo
Page 52 and 53: 46 8. COMPACTNESS weakly convergent
Page 54 and 55: 48 8. COMPACTNESS only if g ∈ L 2
Page 57 and 58: CHAPTER 9 Extension theory We will
Page 59 and 60: 2. SYMMETRIC RELATIONS 53 We will n
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2. SYMMETRIC RELATIONS 55 then it i
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EXERCISES FOR CHAPTER 9 57 Exercise
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60 10. BOUNDARY CONDITIONS the same
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62 10. BOUNDARY CONDITIONS locally
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64 10. BOUNDARY CONDITIONS of a sym
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66 10. BOUNDARY CONDITIONS uniforml
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68 10. BOUNDARY CONDITIONS Hint: If
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70 11. STURM-LIOUVILLE EQUATIONS fu
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72 11. STURM-LIOUVILLE EQUATIONS ob
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74 11. STURM-LIOUVILLE EQUATIONS Ta
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76 11. STURM-LIOUVILLE EQUATIONS Pr
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78 11. STURM-LIOUVILLE EQUATIONS No
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80 11. STURM-LIOUVILLE EQUATIONS Th
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CHAPTER 12 Inverse spectral theory
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1. ASYMPTOTICS OF THE m-FUNCTION 85
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2. UNIQUENESS THEOREMS 87 hand, the
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2. UNIQUENESS THEOREMS 89 the bound
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CHAPTER 13 First order systems We s
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u(c) = 0 is given by (13.2) u(x) =
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13. FIRST ORDER SYSTEMS 95 and (u2,
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13. FIRST ORDER SYSTEMS 97 a non-ze
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EXERCISES FOR CHAPTER 13 99 Exercis
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102 14. EIGENFUNCTION EXPANSIONS We
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104 14. EIGENFUNCTION EXPANSIONS Ex
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106 15. SINGULAR PROBLEMS We obtain
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108 15. SINGULAR PROBLEMS We will g
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110 15. SINGULAR PROBLEMS Proof. Le
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112 15. SINGULAR PROBLEMS in the pr
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114 15. SINGULAR PROBLEMS Since vK
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APPENDIX A Functional analysis In t
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A. FUNCTIONAL ANALYSIS 119 other wo
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122 B. STIELTJES INTEGRALS (4) C (5
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124 B. STIELTJES INTEGRALS Definiti
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126 B. STIELTJES INTEGRALS We obtai
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APPENDIX C Linear first order syste
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C. LINEAR FIRST ORDER SYSTEMS 131 s
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Spectral Theory in Hilbert Space

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