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Partial Differential Equations VII Spectral Theory of Differential Operators

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Partial Differential Equations VII Spectral Theory of Differential Operators M. A. Shubin (Ed.) Partial Differential Equations VII Spectral Theory of Differential Operators Springer-Verlag Berlin Heidelberg Newark London Paris Tokyo Hong Kong Barcelona Budapest Spectral Theory of Differential Operators G.V. Rozenblum, M.A. Shubin, M.Z. Solomyak Translated from the Russian by T. Zastawniak Contents Preface 5 §1 Some Information on the Theory of Operators in a Hilbert Space . 7 1.1. Linear Operators. Closed Operators 7 1.2. The Adjoint Operator 8 1.3. Self-Adjoint Operators 8 1.4. The Spectrum of an Operator 9 1.5. Spectral Measure. The Spectral Theorem for Self-Adjoint Operators 9 1.6. The Pure Point, Absolutely Continuous, and Continuous Singular Components of a Self-Adjoint Operator 11 1.7. Other Formulations of the Spectral Theorem 12 1.8. Semi-Bounded Operators and Forms 13 1.9. The Friedrichs Extension 15 1.10. Variational Triples 15 1.11. The Distribution Function of the Spectrum. The Spectral Function 16 1.12. Compact Operators 18 §2 Defining Differential Operators. Essential Self-Adjointness 19 2.1. Differential Expressions and Their Symbols 19 2.2. Elliptic Differential Expressions 20 2.3. The Maximal and Minimal Operators 21 2 Contents 2.4. Essential Self-Adjointness of Elliptic Operators 23 2.5. Singular Differential Operators 25 2.6. The Schrödinger Operator 26 2.7. The Schrödinger Operator: Local Singularities of the Potential 29 2.8. The Dirac Operator 30 §3 Defining an Operator by a Quadratic Form 31 3.1. Examples 32 3.2. The Schrödinger Operator and Its Generalizations 34 3.3. Non-Semi-Bounded Potentials 35 3.4. Weighted Polyharmonic Operator 36 §4 Examples of Exact Computation of the Spectrum 38 4.1. Operators with Constant Coefficients on Rn and on a Torus . 38 4.2. The Factorization Method 40 4.3. Operators on a Sphere and a Hemisphere 41 §5 Differential Operators with Discrete Spectrum. Estimates of Eigenvalues 42 5.1. Basic Examples of Differential Operators with Discrete Spectrum 43 5.2. Estimates of Eigenvalues 44 5.3. Estimates of the Spectrum of a Weighted Polyharmonic Operator 46 5.4. Estimates of the Spectrum: Heuristic Approach 48 5.5. Estimates of Eigenfunctions 49 §6 Differential Operators with Non-Empty Essential Spectrum 50 6.1. Stability of the Essential Spectrum under Compact Perturbations of the Resolvent 50 6.2. Essential Spectrum of the Schrödinger Operator with Decreasing Potential 51 6.3. Negative Spectrum of the Schrödinger Operator 51 6.4. The Dirac Operator 54 6.5. Eigenvalues within the Continuous Spectrum 55 6.6. On the Essential Spectrum of the Stokes Operator 56 §7 Multiparticle Schrödinger Operator 56 7.1. Definition of the Operator. Centre of Mass Separation 56 7.2. Subsystems. Essential Spectrum 58 7.3. Eigenvalues 60 7.4. Refinement of the Physical Model 61 Contents 3 §8 Investigation of the Spectrum by the Methods of Perturbation Theory 62 8.1. The Rayleigh-Schrödinger Series 63 8.2. Typical Spectral Properties of Elliptic Operators 64 8.3. The Asymptotic Rayleigh-Schrödinger Series 65 8.4. Singular Perturbations 66 8.5. Semiclassical Asymptotics 66 §9 Asymptotic Behaviour of the Spectrum. I. Preliminary Remarks . 68 9.1. Two Forms of Asymptotic Formulae 68 9.2. Formulae for the Leading Term of the Asymptotics 69 9.3. The Weyl Asymptotics for Regular Elliptic Operators 71 9.4. Refinement of the Asymptotic Formulae 74 9.5. Spectrum with Accumulation Point at 0 76 9.6. Semiclassical Asymptotics 77 9.7. Survey of Methods for Obtaining Asymptotic Formulae 78 §10 Asymptotic Behaviour of the Spectrum. II. Operators with 'Non-Weyl' Asymptotics 81 10.1. The General Scheme 81 10.2. The Operator -Дд in Infinite Horn-Shaped Domains 82 10.3. Elliptic Operators Degenerate at the Boundary of the Domain 83 10.4. Hypoelliptic Operators with Double Characteristics 84 10.5. The Cohn-Laplace Operator 85 10.6. The n-Dimensional Schrödinger Operator with Homogeneous Potential 86 10.7. Compact Operators with Non-Weyl Asymptotic Behaviour of the Spectrum 88 §11 Variational Technique in Problems on Spectral Asymptotics 89 11.1. Continuity of Asymptotic Coefficients 89 11.2. Outline of the Proof of Formula (9.25) 90 11.3. Other Applications of the Variational Method 91 11.4. Problems with Constraints 94 §12 The Resolvent and Parabolic Methods. Spectral Geometry 96 12.1. The Resolvent Method 96 12.2. The Case of Non-Weyl Asymptotic Behaviour of the Spectrum 99 12.3. Refinement of the Asymptotic Formulae 100 12.4. The Parabolic Equation Method 101 12.5. Complete Asymptotic Expansion of the ^-Function 103 12.6. Spectral Geometry 104 12.7. Computation of Coefficients 105 4 Contents 12.8. The Problem of Reconstructing the Metric from the Spectrum 106 12.9. Connection with Probability Theory 108 §13 The Hyperbolic Equation Method 108 13.1. Tauberian Theorem for the Fourier Transform 109 13.2. Outline of the Method 112 13.3. Global Fourier Integral Operators 115 13.4. Remarks on Other Problems. Reflection and Branching of Bicharacteristics 121 13.5. Normal Singularity. Two-Term Asymptotic Formulae 126 13.6. Other Results 128 §14 Bicharacteristics and Spectrum 131 14.1. The General Two-Term Asymptotic Formula 132 14.2. Operators with Periodic Bicharacteristic Flow 135 14.3. 'Weak' Non-Zero Singularities of a(t) 137 14.4. Quasimodes 139 14.5. Construction of Quasimodes 140 §15 Approximate Spectral Projection Method 143 15.1. The Basic Concept 143 15.2. Operator Estimates 145 15.3. Construction of an Approximate Spectral Projection 147 15.4. Some Precise Formulations 149 §16 The Laplace Operator on Homogeneous Spaces and on Fundamental Domains of Discrete Groups of Motions 157 16.1. Preliminary Remarks 157 16.2. The Automorphic Laplace Operator 158 16.3. The Laplace Operator on a Flat Torus. The Poisson Formula 158 16.4. The Case of Spaces of Constant Negative Curvature 160 16.5. The Case of Spaces of Constant Positive Curvature 161 16.6. Isospectral Families of Nilmanifolds 164 16.7. Sunada's Technique and Solution of Kac's Problem 165 §17 Operators with Periodic Coefficients 169 17.1. Bloch Functions and the Zone Structure of the Spectrum of an Operator with Periodic Coefficients 169 17.2. The Character of the Spectrum of an Operator with Periodic Coefficients 177 17.3. Quantitative Characteristics of the Spectrum: Global Quasimomentum, Rotation Number, Density of States, and Spectral Function 180 Preface 5 §18 Operators with Almost Periodic Coefficients 186 18.1. General Definitions. Essential Self-Adjointness 186 18.2. General Properties of the Spectrum and Eigenfunctions .... 188 18.3. The Spectrum of the One-Dimensional Schrödinger Operator with an Almost Periodic Potential 192 18.4. The Density of States of an Operator with Almost Periodic Coefficients 197 18.5. Interpretation of the Density of States with the Aid of von Neumann Algebras and Its Properties 199 §19 Operators with Random Coefficients 206 19.1. Translation Homogeneous Random Fields 207 19.2. Random Differential Operators 212 19.3. Essential Self-Adjointness and Spectra 214 19.4. Density of States 217 19.5. The Character of the Spectrum. Anderson Localization 220 §20 Non-Self-Adjoint Differential Operators that Are Close to Self-Adjoint Ones 222 20.1. Preliminary Remarks 222 20.2. Basic Examples 225 20.3. Completeness Theorems 226 20.4. Expansion and Summability Theorems. Asymptotic Behaviour of the Spectrum 228 20.5. Application to Differential Operators 230 Comments on the Literature 234 References 236 Author Index 262 Subject Index 265 .
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