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 Schrodinger Equation. - 2.1. Self-Adjointness. - 2.2. An Estimate of the Growth of Generalized Eigenfunctions. - 2.3. The Schrodinger Operator with Increasing Potential. - 1. Discreteness of spectrum. - 2. Comparison theorems and the behaviour of eigenfunctions as x ? - 3. Theorems on zeros of eigenfunctions. - 2.4. On the Asymptotic Behaviour of Solutions of Certain Second-Order Differential Equations as x ? - 1. The case of integrable potential. - 2. Liouville's transformation and operators with non-integrable potential. - 2.5. On Discrete Energy Levels of an Operator with Semi-Bounded Potential. - 1 The operator in a half-axis with Dirichlet's boundary condition. - 2. The case of an operator on the half-axis with the Neumann boundary condition. - 3. The case of an operator on the whole axis. - 2.6. Eigenfunction Expansion for Operators with Decaying Potentials. - 1. Preliminary remarks. - 2. Formulation of the main theorem. - 3. Two proofs of Theorem 6.1. - 4. One-dimensional oper-ator obtained from the radially symmetric three-dimensional operator. - 5. The case of an operator on the whole axis. - 2.7. The Inverse Problem of Scattering Theory. - 1. Inverse problem on the half-axis. - 2. Inverse problem on the whole axis. - 2.8. Operator with Periodic Potential. - 1. Bloch functions and the band structure of the spectrum. - 2. Expansion into Bloch eigenfunctions. - 3. The density of states. - 3. The Multidimensional Schrodinger Equation. - 3.1. Self-Adjointness. - 3.2. An Estimate of the Generalized Eigenfunctions. - 3.3. Discrete Spectrum and Decay of Eigenfunctions. - 1. Discreteness of spectrum. - 2. Decay of eigenfunctions. - 3. Non-degeneracy of the ground state and positiveness of the first eigenfunction. - 4. On the zeros of eigenfunctions 180. - 3.4. The Schrodinger Operator with Decaying Potential: Essential Spectrum and Eigenvalues. - 1. Essential spectrum. - 2. Separation of variables in the case of spherically symmetric potential and the Laplace-Beltrami operator on a sphere. - 3. Estimation of the number of negative eigenvalues. - 4. Absence of positive eigenvalues. - 3.5. The Schrodinger Operator with Periodic Potential. - 1. Lattices. - 2. Bloch functions. - 3. Expansion in Bloch functions. - 4. Band functions and the band structure of the spectrum. - 5. Theorem on eigenfunction expansion. - 6. Non-triviality of band functions and the absence of a point spectrum. - 7. Density of states. - 4. Scattering Theory. - 4.1. The Wave Operators and the Scattering Operator. - 1. The basic definitions and the statement of the problem. - 2. Physical interpretation. - 3. Properties of the wave operators. - 4. The invariance principle and the abstract conditions for the existence and completeness of the wave operators. - 4.2. Existence and Completeness of the Wave Operators. - 1. The abstract scheme of Enss. - 2. The case of the Schrodinger operator. - 3. The scattering matrix. - 4. One-dimensional case. - 5. Spherically symmetric case. - 4.3. The Lippman-Schwinger Equations and the Asymptotics of Eigen-functions. - 1. A derivation of the Lippman-Schwinger equations. - 2. Another derivation of the Lippman-Schwinger equations. - 3. An outline of the proof of the completeness of wave operators by the stationary method. - 4. Discussion on the Lippman-Schwinger equation. - 5. Asymptotics of eigenfunctions. - 5. Symbols of Operators and Feynman Path Integrals. - 5.1. Symbols of Operators and Quantization: qp-and pq-Symbols and Weyl Symbols. - 1. The general concept of symbol and its connection with quantization. - 2. The qp-and pq-symbols. - 3. Symmetric or Weyl symbols. - 4. Weyl symbols and linear canonical transformations. - 5. Weyl symbols and reflections. - 5.2. Wick and Anti-Wick Symbols. Covariant and Contravariant Symbols. - 1. Annihilation and creation operators. Fock space. - 2. Definition and elementary properties of Wick and Anti-Wick symbols. - 3. Covariant and contravariant symbols. - 4. Convexity inequalities and Feynman-type inequalities. - 5.3. The General Concept of Feynman Path Integral in Phase Space. Symbols of the Evolution Operator. - 1. The method of Feynman Path integrals. - 2. Weyl symbol of the evolution operator. - 3. The Wick symbol of the evolution operator. - 4. pq-and qp-symbols of the evolution operator and the path integral for matrix elements. - 5.4. Path Integrals for the Symbol of the Scattering Operator and for the Partition Function. - 1. Path integral for the symbol of the scattering operator. - 2. The path integral for the partition function. - 5.5. The Connection between Quantum and Classical Mechanics. Semiclassical Asymptotics. - 1. The concept of a semiclassical asymptotic. - 2. The operator initial-value problem. - 3. Asymptotics of the Green's function. - 4. Asymptotic behaviour of eigenvalues. - 5. Bohr's formula 383. - Supplement 1. Spectral Theory of Operators in Hilbert Space. - S1.1. Operators in Hilbert Space. The Spectral Theorem. - 1. Preliminaries. - 2. Theorem on the spectral decomposition of a self-adjoint operator in a separable Hilbert space. - 3. Examples and exercises. - 4. Commuting self-adjoint operators in Hilbert space, operators with simple spectrum. - 5. Functions of self-adjoint operators. - 6. One-parameter groups of unitary operators. - 7. Operators with simple spectrum. - 8. The classification of spectra. - 9. Problems and exercises. - S1.2. Generalized Eigenfunctions. - 1. Preliminary remarks. - 2. Hilbert-Schmidt operators. - 3. Rigged Hilbert spaces. - 4. Generalized eigenfunctions. - 5. Statement and proof of main theorem. - 6. Appendix to the main theorem. - 7. Generalized eigenfunctions of differential operators. - S1.3. Variational Principles and Perturbation Theory for a Discrete Spectrum. - S1.4. Trace Class Operators and the Trace. - 1. Definition and main properties. - 2. Polar decomposition of an operator. - 3. Trace norm. - 4. Expressing the trace in terms of the kernel of the operator. - 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. - 1. Superspaces and supermanifolds. - 2. Classical Lie superalgebras. - 3. Lie supergroups and homogeneous superspaces in ternis of the point functor. - 4. Two types of mechanics on supermanifolds and Shander's time. - S3.2. Quantization: main procedures. - S3.3. Supersymmetry of the Ordinary Schrodinger Equation and of the Electron in the Non-Homogeneous Magnetic Field. - A Short Guide to the Bibliography.