This volume deals with those topics of mathematical physics, associated with the study of the Schr”dinger equation, which are considered to be the most important.
Chapter 1 presents the basic concepts of quantum mechanics. Chapter 2 provides an introduction to the spectral theory of the one-dimensional Schr”dinger equation. Chapter 3 opens with a discussion of the spectral theory of the multi-dimensional Schr”dinger equation, which is a far more complex case and requires careful consideration of aspects which are trivial in the one-dimensional case. Chapter 4 presents the scattering theory for the multi-dimensional non-relativistic Schr”dinger equation, and the final chapter is devoted to quantization and Feynman path integrals. These five main chapters are followed by three supplements, which present material drawn on in the various chapters. The first two supplements deal with general questions concerning the spectral theory of operators in Hilbert space, and necessary information relating to Sobolev spaces and elliptic equations. Supplement 3, which essentially stands alone, introduces the concept of the supermanifold which leads to a more natural treatment of quantization.
Although written primarily for mathematicians who wish to gain a better awareness of the physical aspects of quantum mechanics and related topics, it will also be useful for mathematical physicists who wish to become better acquainted with the mathematical formalism of quantum mechanics. Much of the material included here has been based on lectures given by the authors at Moscow State University, and this volume can also be recommended as a supplementary graduate level introduction to the spectral theory of differential operators with both discrete and continuous spectra.
This English edition is a revised, expanded version of the original Soviet publication.
' ....interest to both mathematicians and physicists; Highly recommended.' Mathematika 39 1992
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.