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Berezin F.A., Shubin M.A. The Schrodinger Equation
Kluwer Academic Publishers ,1991. - 555 p.
The Schrodinger equation is the basic equation of quantum theory. The study of this equation plays an exceptionally important role in mode physics. From a mathematician's point of view the Schrodinger equation is as inexhaustible as mathematics itself.
In this book an attempt has been made to set forth those topics of mathematical physics, associated with the study of the Schr/Sdinger equation, which appear to be the most important.
Intended mainly for students of mathematics, the book starts with an introductory chapter dealing with the basic concepts of quantum mechanics. This would help the reader well versed in mathematics to understand the physical meaning of the mathematical constructions and theorems expounded in the subsequent chapters. One should not think that this concise chapter can serve
as a substitute for a systematic study of physical textbooks on quantum mechanics. It is hoped, however, that the perusal of the book would be sufficient for a mathematician to take in these textbooks.
This book is based on the lectures given more than once by the authors at the Department of Mechanics and Mathematics of the Moscow State University and in part published in 1972. Though the lectures were meant for students of mathematics, the experience with the above mentioned preliminary publication showed that the type of material presented here can also be useful to the physicists who want to familiarize themselves better with the mathematical formalism of quantum mechanics.


Series Editor's Preface


Notational Conventions

General Concepts of Quantum Mechanics

Formulation of Basic Postulates
Some Corollaries of the Basic Postulates
Time Differentiation of Observables
The Uncertainty Relations and Simultaneous Measurability of Physical Quantities
The Free Particle in Three-Dimensional Space
Particles with Spin
Harmonic Oscillator
Identical Particles
Second Quantization

The One-Dimensional Schrodinger Equation

An Estimate of the Growth of Generalized Eigenfunctions
The Schrodinger Operator with Increasing Potential

Discreteness of spectrum
Comparison theorems and the behaviour of eigenfunctions
Theorems on zeros of eigenfunctions
On the Asymptotic Behaviour of Solutions of Certain Second-Order Differential Equations
The case of integrable potential
Liouville's transformation and operators with non-integrable potential
On Discrete Energy Levels of an Operator with Semi-Bounded Potential
The operator in a half-axis with Dirichlet's boundary condition
The case of an operator on the half-axis with the Neumann boundary condition
The case of an operator on the whole axis
Eigenfunction Expansion for Operators with Decaying Potentials
Preliminary remarks
Formulation of the main theorem
Two proofs of Theorem 6.1.
One-dimensional operator obtained from the radially symmetric three-dimensional operator
The case of an operator on the whole axis
The Inverse Problem of Scattering Theory
Inverse problem on the half-axis
Inverse problem on the whole axis
Operator with Periodic Potential
Bloch functions and the band structure of the spectrum
Expansion into Bloch eigenfunctions
The density of states

The Multidimensional SchrSdinger Equation

An Estimate of the Generalized Eigenfunctions
Discrete Spectrum and Decay of Eigenfunctions

Discreteness of spectrum
Decay of eigenfunctions
Non-degeneracy of the ground state and positivehess of the first eigenfunction
On the zeros of eigenfunctions
The SchrSdinger Operator with Decaying Potential: Essential Spectrum and Eigenvalues
Essential spectrum
Separation of variables in the case of spherically symmetric potential and the Laplace-Beltrami operator on a sphere
Estimation of the number of negative eigenvalues
Absence of positive eigenvalues
The Schrodinger Operator with Periodic Potential
Bloch functions
Expansion in Bloch functions
Band functions and the band structure of the spectrum
Theorem on eigenfunction expansion
Non-triviality of band functions and the absence of a point spectrum
Density of states

Scattering Theory

The Wave Operators and the Scattering Operator
The basic definitions and the statement of the problem
Physical interpretation
Properties of the wave operators
The invariance principle and the abstract conditions for the existence and completeness of the wave operators
Existence and Completeness of the Wave Operators
The abstract scheme of Enss
The case of the Schrodinger operator
The scattering matrix
One-dimensional case
Spherically symmetric case
The Lippman-Schwinger Equations and the Asymptotics of Eigen-functions
A derivation of the Lippman-Schwinger equations
Another derivation of the Lippman-Schwinger equations
An outline of the proof of the completeness of wave operators by the stationary method
Discussion on the Lippman-Schwinger equation
Asymptotics of eigenfunctions

Symbols of Operators and Feynman Path Integrals

Symbols of Operators and Quantization: qp- and pq-Symbols and Weyl Symbols
The general concept of symbol and its connection with quantization
The qp- and pq-symbols
Symmetric or Weyl symbols
Weyl symbols and linear canonical transformations
Weyl symbols and reflections
Wick and Anti-Wick Symbols. Covariant and Contravariant Symbols
Annihilation and creation operators. Fock space
Definition and elementary properties of Wick and Anti-Wick symbols
Covariant and contravariant symbols
Convexity inequalities and Feynman-type inequalities
The General Concept of Feynman Path Integral in Phase Space. Symbols of the Evolution Operator
The method of Feynman Path integrals
Weyl symbol of the evolution operator
The Wick symbol of the evolution operator
pq- and qp-symbols of the evolution operator and the path integral for matrix elements
Path Integrals for the Symbol of the Scattering Operator and for the Partition Function
Path integral for the symbol of the scattering operator
The path integral for the partition function
The Connection between Quantum and Classical Mechanics. Semiclassical Asymptotics
The concept of a semiclassical asymptotic
The operator initial-value problem
Asymptotics of the Green's function
Asymptotic behaviour of eigenvalues
Bohr's formula


Spectral Theory of Operators in Hilbert Space

Operators in ttilbert Space. The Spectral Theorem
Theorem on the spectral decomposition of a self-adjoint operator in a separable Hilbert space
Examples and exercises
Commuting self-adjoint operators in Hilbert space, operators with simple spectrum
Functions of self-adjoint operators
One-parameter groups of unitary operators
Operators with simple spectrum
The classification of spectra
Problems and exercises
Generalized Eigenfunctions
Preliminary remarks
Hilbert-Schmidt operators
Rigged Hilbert spaces
Generalized eigenfunctions
Statement and proof of main theorem
Appendix to the main theorem
Generalized eigenfunctions of differential operators
Variational Principles and Perturbation Theory for a Discrete Spectrum
Trace Class Operators and the Trace
Definition and main properties
Polar decomposition of an operator
Trace norm
Expressing the trace in terms of the keel of the operator
Tensor Products of Hilbert Spaces

Sobolev Spaces and Elliptic Equations

Sobolev Spaces and Embedding Theorems
Regularity of Solutions of Elliptic Equations and a priori Estimates
Singularities of Green's Functions

Quantization and Supermanifolds

Supermanifolds: Recapitulations
Superspaces and supermanifolds
Classical Lie superalgebras
Lie supergroups and homogeneous superspaces in terms of the point functor
Two types of mechanics on supermanifolds and Shander's time
Quantization: main procedures
Supersymmetry of the Ordinary Schrodinger Equation and of the Electron in the Non-Homogeneous Magnetic Field

A Short Guide to the Bibliography
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