Spectral Theory - Basic Concepts and Applications

Description

This textbook offers a concise introduction to spectral theory, designed for newcomers to functional analysis. Curating the content carefully, the author builds to a proof of the spectral theorem in the early part of the book. Subsequent chapters illustrate a variety of application areas, exploring key examples in detail. Readers looking to delve further into specialized topics will find ample references to classic and recent literature. Beginning with a brief introduction to functional analysis, the text focuses on unbounded operators and separable Hilbert spaces as the essential tools needed for the subsequent theory. A thorough discussion of the concepts of spectrum and resolvent follows, leading to a complete proof of the spectral theorem for unbounded self-adjoint operators. Applications of spectral theory to differential operators comprise the remaining four chapters. These chapters introduce the Dirichlet Laplacian operator, Schrödinger operators, operators on graphs, and the spectral theory of Riemannian manifolds. Spectral Theory offers a uniquely accessible introduction to ideas that invite further study in any number of different directions. A background in real and complex analysis is assumed; the author presents the requisite tools from functional analysis within the text. This introductory treatment would suit a functional analysis course intended as a pathway to linear PDE theory. Independent later chapters allow for flexibility in selecting applications to suit specific interests within a one-semester course. Preface Contents 1 Introduction Notes 2 Hilbert Spaces 2.1 Normed Vector Spaces 2.2 Lp Spaces 2.3 Bounded Linear Maps 2.3.1 Operator Topologies 2.3.2 Uniform Boundedness 2.4 Hilbert Spaces 2.5 Sobolev Spaces 2.5.1 Weak Derivatives 2.5.2 Hm Spaces 2.6 Orthogonality 2.7 Orthonormal Bases 2.7.1 Weak Sequential Compactness 2.8 Exercises Notes 3 Operators 3.1 Unbounded Operators 3.2 Adjoints 3.2.1 Adjoints of Unbounded Operators 3.3 Closed Operators 3.3.1 Closable Operators 3.3.2 Closed Graph Theorem 3.3.3 Invertibility 3.4 Symmetry and Self-adjointness 3.4.1 Self-adjoint Operators 3.4.2 Criteria for Self-adjointness 3.4.3 Friedrichs Extension 3.5 Compact Operators 3.5.1 Hilbert–Schmidt Operators 3.6 Exercises Notes 4 Spectrum and Resolvent 4.1 Definitions and Examples 4.1.1 Basic Properties of the Spectrum 4.1.2 Spectrum of a Multiplication Operator 4.1.3 Resolvent of the Euclidean Laplacian 4.1.4 Discrete Laplacians 4.2 Resolvent 4.2.1 Analytic Operator-Valued Functions 4.2.2 Analyticity of the Resolvent 4.2.3 Spectral Radius 4.3 Spectrum of Self-adjoint Operators 4.4 Spectral Theory of Compact Operators 4.4.1 Spectral Theorem for Compact Self-adjoint Operators 4.4.2 Hilbert–Schmidt Operators 4.4.3 Traces 4.5 Exercises Notes 5 The Spectral Theorem 5.1 Unitary Operators 5.1.1 Continuous Functional Calculus 5.1.2 Spectral Measures 5.1.3 Spectral Theorem for Unitary Operators 5.2 The Main Theorem 5.3 Functional Calculus 5.4 Spectral Decomposition 5.4.1 Discrete and Essential Spectrum 5.4.2 Continuous Spectrum 5.4.3 The Min–Max Principle 5.5 Exercises Notes 6 The Laplacian with Boundary Conditions 6.1 Self-adjoint Extensions 6.1.1 The Space H10(Ω) 6.1.2 The Dirichlet Laplacian 6.1.3 The Neumann Laplacian 6.2 Discreteness of Spectrum 6.2.1 Periodic Sobolev Spaces 6.2.2 Extension Lemmas 6.3 Regularity of Eigenfunctions 6.4 Eigenvalue Computations 6.4.1 Finite Element Method 6.4.2 Domain Monotonicity 6.4.3 Neumann Eigenvalues 6.5 Asymptotics of Dirichlet Eigenvalues 6.5.1 Strategy for the Proof 6.5.2 Asymptotics of the Resolvent Kernel 6.5.3 Trace Asymptotics 6.5.4 The Tauberian Argument 6.6 Nodal Domains 6.7 Isoperimetric Inequalities and Minimal Eigenvalues 6.8 Exercises Notes 7 Schrödinger Operators 7.1 Positive Potentials 7.1.1 Essential Self-adjointness 7.1.2 Quadratic Form Extension 7.1.3 Discrete Spectrum 7.1.4 Quantum Harmonic Oscillator 7.2 Relatively Bounded Perturbations 7.3 Relatively Compact Perturbations 7.4 Hydrogen Atom 7.5 Semiclassical Asymptotics 7.6 Periodic Potentials 7.6.1 Floquet Theory 7.6.2 Spectrum of H 7.7 Exercises Notes 8 Operators on Graphs 8.1 Combinatorial Laplacians 8.2 Quantum Graphs 8.3 Spectral Properties of Compact Quantum Graphs 8.4 Eigenvalue Comparison 8.5 Eigenvalue Asymptotics 8.5.1 Weyl Law 8.6 Exercises Notes 9 Spectral Theory on Manifolds 9.1 Smooth Manifolds 9.1.1 Tangent and Cotangent Vectors 9.1.2 Partition of Unity 9.2 Riemannian Metrics 9.2.1 Geodesics and the Exponential Map 9.2.2 Completeness 9.3 The Laplacian 9.3.1 Green's Identity 9.4 Spectrum of a Compact Manifold 9.4.1 Dirichlet Eigenvalues 9.4.2 Regularity 9.5 Heat Equation 9.5.1 Maximum Principle 9.5.2 Heat Kernel 9.5.3 Spectral Applications 9.6 Wave Propagation on Compact Manifolds 9.6.1 Propagation Speed 9.7 Complete Manifolds and Essential Self-adjointness 9.8 Essential Spectrum of Complete Manifolds 9.8.1 Decomposition Principle 9.8.2 The Bottom of the Essential Spectrum 9.8.3 Volume Growth Estimate 9.9 Exercises Notes A Background Material A.1 Measure and Integration A.1.1 Lebesgue Measure A.1.2 Integration A.1.3 Product Measure A.1.4 Differentiation A.1.5 Decomposition of Measures A.1.6 Riesz Representation A.2 Lp Spaces A.2.1 Completeness A.2.2 Convolution A.3 Fourier Transform A.4 Elliptic Regularity References Index