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 SobolevSpaces and Convolutions. - A4. 1 Fourier Transform. - A4. 2 Sobolev Spaces. - A4. 3 Convolutions. - References. Language: English