Introduction to Spectral Theory

1 The Spectrum of Linear Operators and Hilbert Spaces.- 2 The Geometry of a Hilbert Space and Its Subspaces.- 3 Exponential Decay of Eigenfunctions.- 4 Operators on Hilbert Spaces.- 5 Self-Adjoint Operators.- 6 Riesz Projections and Isolated Points of the Spectrum.- 7 The Essential Spectrum: Weyl's Criterion.- 8 Self-Adjointness: Part 1. The Kato Inequality.- 9 Compact Operators.- 10 Locally Compact Operators and Their Application to Schrödinger Operators.- 11 Semiclassical Analysis of Schrödinger Operators I: The Harmonic Approximation.- 12 Semiclassical Analysis of Schrödinger Operators II: The Splitting of Eigenvalues.- 13 Self-Adjointness: Part 2. The Kato-Rellich Theorem 131.- 14 Relatively Compact Operators and the Weyl Theorem.- 15 Perturbation Theory: Relatively Bounded Perturbations.- 16 Theory of Quantum Resonances I: The Aguilar-Balslev-Combes-Simon Theorem.- 17 Spectral Deformation Theory.- 18 Spectral Deformation of Schrödinger Operators.- 19 The General Theory of Spectral Stability.- 20 Theory of Quantum Resonances II: The Shape Resonance Model.- 21 Quantum Nontrapping Estimates.- 22 Theory of Quantum Resonances III: Resonance Width.- 23 Other Topics in the Theory of Quantum Resonances.- Appendix 1. Introduction to Banach Spaces.- A1.1 Linear Vector Spaces and Norms.- A1.2 Elementary Topology in Normed Vector Spaces.- A1.3 Banach Spaces.- A1.4 Compactness.- 1. Density results.- 2. The Hölder Inequality.- 3. The Minkowski Inequality.- 4. Lebesgue Dominated Convergence.- Appendix 3. Linear Operators on Banach Spaces.- A3.1 Linear Operators.- A3.2 Continuity and Boundedness of Linear Operators.- A3.3 The Graph of an Operator and Closure.- A3.4 Inverses of Linear Operators.- A3.5 Different Topologies on L(X).- Appendix 4. The Fourier Transform, Sobolev Spaces, and Convolutions.- A4.1 Fourier Transform.- A4.2 Sobolev Spaces.- A4.3 Convolutions.- References.