Introduction to Spectral Theory

Transcription

1 P.D. Hislop I.M. Sigal Introduction to Spectral Theory With Applications to Schrodinger Operators Springer

2 Introduction and Overview 1 1 The Spectrum of Linear Operators and Hilbert Spaces TheSpectrum Properties of the Resolvent Hilbert Space 13 2 The Geometry of a Hilbert Space and Its Subspaces Subspaces Linear Functionals and the Riesz Theorem \ Orthonormal Bases 22 3 Exponential Decay of Eigenfunctions Introduction Agmbn Metric The Main Theorem Proof of Theorem Pointwise Exponential Bounds Notes? 37

3 vi 4 Operators on Hilbert Spaces Remarks on the Operator Norm and Graphs The Adjoint of an Operator Unitary Operators 45 5 Self-Adjoint Operators Definitions General Properties of Self-Adjoint Operators Determining the Spectrum of Self-Adjoint Operators Projections 56 6 Riesz Projections and Isolated Points of the Spectrum Riesz Projections Isolated Points of the Spectrum More Properties of Riesz Projections Embedded Eigenvalues of Self-Adjoint Operators 67 7 The Essential Spectrum: Weyl's Criterion The Weyl Criterion Proof of Weyl's Criterion: First Part Proof of Weyl's Criterion: Second Part 72 8 Self-Adjointness: Part 1. The Kato Inequality Symmetric Operators Fundamental Criteria for Self-Adjointness The Kato Inequality for Smooth Functions Technical Approximation Tools The Kato Inequality Application to Positive Potentials 85 9 Compact Operators ; Compact and Finite-Rank Operators ; The Structure of the Set of Compact Operators Spectral Theory of Compact Operators Applications of the General Theory Locally Compact Operators and Their Application to Schrodinger Operators Locally Compact Operators Spectral Properties of Locally Compact Operators Essential Spectrum and Weyl's Criterion for Certain Closed Operators Semiclassical Analysis of Schrodinger Operators I: The Harmonic Approximation Introduction 109

4 vii 11.2 Preliminary: The Harmonic Oscillator Semiclassical Limit of Eigenvalues Notes Semiclassical Analysis of Schrodinger Operators II: The Splitting of Eigenvalues More Spectral Analysis: Variational Inequalities Double-Well Potentials and Tunneling Proof of Theorem Appendix: Exponential Decay of Eigenfunctions for Double-Well Hamiltonians Notes Self-Adjointness: Part 2. The Kato-Rellich Theorem Relatively Bounded Operators Schrodinger Operators with Relatively Bounded Potentials Relatively Compact Operators and the Weyl Theorem Relatively Compact Operators Weyl's Theorem: Stability of the Essential Spectrum Applications to the Spectral Theory of Schrodinger Operators Persson's Theorem: The Bottom of the Essential Spectrum Perturbation Theory: Relatively Bounded Perturbations Introduction and Motivation Analytic Perturbation Theory for the Discrete Spectrum Criteria for Eigenvalue Stability: A Simple Case Type-A Families of Operators and Eigenvalue Stability: General Results Remarks on Perturbation Expansions Appendix: A Technical Lemma ' Theory of Quantum Resonances I: The Aguilar-Balslev-Combes-Simon Theorem Introduction to Quantum Resonance Theory Aguilar-Balslev-Combes-Simon Theory of Resonances Proof of the Aguilar-Balslev-Combes Theorem Examples of the Generalized Semiclassical Regime Notes Spectral Deformation Theory Introduction to Spectral Deformation Vector Fields and Diffeomorphisms Induced Unitary Operators Complex Extensions and Analytic Vectors Notes 186

5 viii 18 Spectral Deformation of Schrodinger Operators The Deformed Family of Schrodinger Operators The Spectrum of the Deformed Laplacian Admissible Potentials J The Spectrum of Deformed Schrodinger Operators Notes The General Theory of Spectral Stability Examples of Nonanalytic Perturbations Strong Resolvent Convergence The General Notion of Stability A Criterion for Stability Proof of the Stability Criteria Geometric Techniques and Applications to Stability Example: A Simple Shape Resonance Model Theory of Quantum Resonances II: The Shape Resonance Model Introduction: The Gamow Model of Alpha Decay The Shape Resonance Model The Semiclassical Regime and Scaling Analyticity Conditions on the Potential Spectral Stability for Shape Resonances: The Main Results The Proof of Spectral Stability for Shape Resonances Resolvent Estimates for H { (k,6) and H(k, 6) Notes Quantum Nontrapping Estimates Introduction to Quantum Nontrapping..." The Classical Nontrapping Condition The Nontrapping Resolvent Estimate > Some examples of Nontrapping Potentials Notes Theory of Quantum Resonances III: Resonance Width Introduction and Geometric Preliminaries... r. ' Exponential Decay of Eigenfunctions of HQ(X) The Proof of Estimates on Resonance Positions Other Topics in the Theory of Quantum Resonances Stark and Stark Ladder Resonances Resonances and the Zeeman Effect Resonances of the Helmholtz Resonator Comments on More General Potentials, Exponential Decay, and Lower Bounds 280

6 ix Appendix 1. Introduction to Banach Spaces 285 Al.l Linear Vector Spaces and Norms 285 A 1.2 Elementary Topology in Normed Vector Spaces 286 A1.3 Banach Spaces 288 A1.4 Compactness 291 Appendix 2. The Banach Spaces LP(IR"), 1 < p < oo 293 A2.1 The Definition of LP(/R"), 1 < p < oo 293 A2.2 Important Properties of L p -Spaces Density results The Holder Inequality The Minkowski Inequality Lebesgue Dominated Convergence 299 Appendix 3. Linear Operators on Banach Spaces 301 A3.1 Linear Operators 301 A3.2 Continuity and Boundedness of Linear Operators 303 A3.3 The Graph of an Operator and Closure 307 A3.4 Inverses of Linear Operators 309 A3.5 Different Topologies on C(X) 312 Appendix 4. The Fourier Transform, Sobolev Spaces, and Convolutions 313 A4.1 Fourier Transform 313 A4.2 Sobolev Spaces 316 A4.3 Convolutions 317 References 319 Index 333