The Schrödinger Equation

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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.