1 Vector spaces with a scalar product pre-Hilbert spaces. - 1. 1 Sesquilinear forms. - 1. 2 Scalar products and norms. - 2 Hilbert spaces. - 2. 1 Convergence and completeness. - 2. 2 Topological notions. - 3 Orthogonality. - 3. 1 The projection theorem. - 3. 2 Orthonormal systems and orthonormal bases. - 3. 3 Existence of orthonormal bases dimension of a Hilbert space. - 3. 4 Tensor products of Hilbert spaces. - 4 Linear operators and their adjoints. - 4. 1 Basic notions. - 4. 2 Bounded linear operators and functionals. - 4. 3 Isomorphisms completion. - 4. 4 Adjoint operator. - 4. 5 The theorem of Banach-Steinhaus strong and weak convergence. - 4. 6 Orthogonal projections isometric and unitary operators. - 5 Closed linear operators. - 5. 1 Closed and closable operators the closed graph theorem. - 5. 2 The fundamentals of spectral theory. - 5. 3 Symmetric and self-adjoint operators. - 5. 4 Self-adjoint extensions of symmetric operators. - 5. 5 Operators defined by sesquilinear forms (Friedrichs' extension). - 5. 6 Normal operators. - 6 Special classes of linear operators. - 6. 1 Finite rank and compact operators. - 6. 2 Hilbert-Schmidt operators and Carleman operators. - 6. 3 Matrix operators and integral operators. - 6. 4 Differential operators on L2(a b) with constant coefficients. - 7 The spectral theory of self-adjoint and normal operators. - 7. 1 The spectral theorem for compact operators the spaces Bp (H1H2). - 7. 2 Integration with respect to a spectral family. - 7. 3 The spectral theorem for self-adjoint operators. - 7. 4 Spectra of self-adjoint operators. - 7. 5 The spectral theorem for normal operators. - 7. 6 One-parameter unitary groups. - 8 Self-adjoint extensions of symmetric operators. - 8. 1 Defect indices and Cayley transforms. - 8. 2 Construction of self-adjoint extensions. - 8. 3 Spectra of self-adjoint extensionsof a symmetric operator. - 8. 4 Second order ordinary differential operators. - 8. 5 Analytic vectors and tensor products of self-adjoint operators. - 9 Perturbation theory for self-adjoint operators. - 9. 1 Relatively bounded perturbations. - 9. 2 Relatively compact perturbations and the essential spectrum. - 9. 3 Strong resolvent convergence. - 10 Differential operators on L2(?m). - 10. 1 The Fourier transformation on L2(?m). - 10. 2 Sobolev spaces and differential operators on L2(?m) with constant coefficients. - 10. 3 Relatively bounded and relatively compact perturbations. - 10. 4 Essentially self-adjoint Schrödinger operators. - 10. 5 Spectra of Schrödinger operators. - 10. 6 Dirac operators. - 11 Scattering theory. - 11. 1 Wave operators. - 11. 2 The existence and completeness of wave operators. - 11. 3 Applications to differential operators on L2(?m). - A. 1 Definition of the integral. - A. 2 Limit theorems. - A. 3 Measurable functions and sets. - A. 4 The Fubini-Tonelli theorem. - A. 5 The Radon-Nikodym theorem. - References. - Index of symbols. - Author and subject index. Language: English