The Schroedinger Equation by F.A. Berezin

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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 Schroedinger Equation.- 2.1. Self-Adjointness.- 2.2. An Estimate of the Growth of Generalized Eigenfunctions.- 2.3. The Schroedinger 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 Schroedinger 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 Schroedinger 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 Schroedinger 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 Schroedinger 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 Schroedinger Equation and of the Electron in the Non-Homogeneous Magnetic Field.- A Short Guide to the Bibliography.

The Schroedinger Equation F.A. Berezin

The Schroedinger Equation by F.A. Berezin

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The Schroedinger Equation Summary

The Schroedinger Equation by F.A. Berezin

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