The Schrödinger Equation

4Et moi, ..., si j'avait su comment en revenir, One service mathematics has rendered the je n'y serais point alle.' human race. It has put common sense back Jules Verne where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded non· The series is divergent; therefore we may be sense'. able to do something withit. Eric T. Bell O. Heaviside Mathematicsis a tool for thought. A highly necessary tool in a world whereboth feedback and non linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics ...'; 'One service logic has rendered com puter science ...'; 'One service category theory has rendered mathematics ...'. All arguably true. And all statements obtainable this way form part of the raison d' tre ofthis series.

Klappentext

4Et moi, ..., si j'avait su comment en revenir, One service mathematics has rendered the je n'y serais point alle.' human race. It has put common sense back Jules Verne where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded non· The series is divergent; therefore we may be sense'. able to do something withit. Eric T. Bell O. Heaviside Mathematicsis a tool for thought. A highly necessary tool in a world whereboth feedback and non­ linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics ...'; 'One service logic has rendered com­ puter science ...'; 'One service category theory has rendered mathematics ...'. All arguably true. And all statements obtainable this way form part of the raison d' tre ofthis series.

Zusammenfassung
' ....interest to both mathematicians and physicists; Highly recommended.' Mathematika 39 1992

Inhalt

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.