The Schrödinger Equation

1. General Concepts of Quantum Mechanics.- 1.1. Formulation of Basic Postulates.- 1.2. Some Corollaries of the Basic Postulates.- 1.3. Time Differentiation of Observables.- 1.4. Quantization.- 1.5. The Uncertainty Relations and Simultaneous Measurability of Physical Quantities.- 1.6. The Free Particle in Three-Dimensional Space.- 1.7. Particles with Spin.- 1.8. Harmonic Oscillator.- 1.9. Identical Particles.- 1.10. Second Quantization.- 2. The One-Dimensional Schrödinger Equation.- 2.1. Self-Adjointness.- 2.2. An Estimate of the Growth of Generalized Eigenfunctions.- 2.3. The Schrödinger Operator with Increasing Potential.- 2.4. On the Asymptotic Behaviour of Solutions of Certain Second-Order Differential Equations as x ??.- 2.5. On Discrete Energy Levels of an Operator with Semi-Bounded Potential.- 2.6. Eigenfunction Expansion for Operators with Decaying Potentials...- 2.7. The Inverse Problem of Scattering Theory.- 2.8. Operator with Periodic Potential.- 3. The Multidimensional Schrödinger Equation.- 3.1. Self-Adjointness.- 3.2. An Estimate of the Generalized Eigenfunctions.- 3.3. Discrete Spectrum and Decay of Eigenfunctions.- 3.4. The Schrödinger Operator with Decaying Potential: Essential Spectrum and Eigenvalues.- 3.5. The Schrödinger Operator with Periodic Potential.- 4. Scattering Theory.- 4.1. The Wave Operators and the Scattering Operator.- 4.2. Existence and Completeness of the Wave Operators.- 4.3. The Lippman-Schwinger Equations and the Asymptotics of Eigen-functions.- 5. Symbols of Operators and Feynman Path Integrals.- 5.1. Symbols of Operators and Quantization: qp-and pq-Symbols and Weyl Symbols.- 5.2. Wick and Anti-Wick Symbols. Covariant and Contravariant Symbols.- 5.3. The General Concept of Feynman Path Integral in Phase Space. Symbols ofthe Evolution Operator.- 5.4. Path Integrals for the Symbol of the Scattering Operator and for the Partition Function.- 5.5. The Connection between Quantum and Classical Mechanics. Semiclassical Asymptotics.- Supplement 1. Spectral Theory of Operators in Hilbert Space.- S1.1. Operators in Hilbert Space. The Spectral Theorem.- S1.2. Generalized Eigenfunctions.- S1.3. Variational Principles and Perturbation Theory for a Discrete Spectrum.- S1.4. Trace Class Operators and the Trace.- S1.5. Tensor Products of Hilbert Spaces.- Supplement 2. Sobolev Spaces and Elliptic Equations.- S2.1. Sobolev Spaces and Embedding Theorems.- S2.2. Regularity of Solutions of Elliptic Equations and a priori Estimates.- S2.3. Singularities of Green’s Functions.- Supplement 3. Quantization and Supermanifolds.- S3.1.Supermanifolds:Recapitulations.- S3.2. Quantization: main procedures.- S3.3. Supersymmetry of the Ordinary Schrödinger Equation and of the Electron in the Non-Homogeneous Magnetic Field.- A Short Guide to the Bibliography.