Solvable Models in Quantum Mechanics

Next to the harmonic oscillator and the Coulomb potential the class of two-body models with point interactions is the only one where complete solutions are available. All mathematical and physical quantities can be calculated explicitly which makes this field of research important also for more complicated and realistic models in quantum mechanics. The detailed results allow their implementation in numerical codes to analyse properties of alloys, impurities, crystals and other features in solid state quantum physics. This monograph presents in a systematic way the mathematical approach and unifies results obtained in recent years. The student with a sound background in mathematics will get a deeper understanding of Schrödinger Operators and will see many examples which may eventually be used with profit in courses on quantum mechanics and solid state physics. The book has textbook potential in mathematical physics and is suitable for additional reading in various fields of theoretical quantum physics.

I The One-Center Point Interaction.- I.1 The One-Center Point Interaction in Three Dimensions.- I.2 Coulomb Plus One-Center Point Interaction in Three Dimensions.- I.3 The One-Center ?-Interaction in One Dimension.- I.4 The One-Center ??-Interaction in One Dimension.- I.5 The One-Center Point Interaction in Two Dimensions.- II Point Interactions with a Finite Number of Centers.- II.1 Finitely Many Point Interactions in Three Dimensions.- II.2 Finitely Many ?-Interactions in One Dimension.- II.3 Finitely Many ??-Interactions in One Dimension.- II.4 Finitely Many Point Interactions in Two Dimensions.- III Point Interactions with Infinitely Many Centers.- III.1 Infinitely Many Point Interactions in Three Dimensions.- III.2 Infinitely Many ?-Interactions in One Dimension.- III.3 Infinitely Many ??-Interactions in One Dimension.- III.4 Infinitely Many Point Interactions in Two Dimensions.- III.5 Random Hamiltonians with Point Interactions.- Appendices.- A Self-Adjoint Extensions of Symmetric Operators.- B Spectral Properties of Hamiltonians Defined as Quadratic Forms.- C Schrödinger Operators with Interactions Concentrated Around Infinitely Many Centers.- D Boundary Conditions for Schrödinger Operators on (0, ?).- E Time-Dependent Scattering Theory for Point Interactions.- F Dirichlet Forms for Point Interactions.- G Point Interactions and Scales of Hilbert Spaces.- H Nonstandard Analysis and Point Interactions.- H.1 A Very Short Introduction to Nonstandard Analysis.- H.2 Point Interactions Using Nonstandard Analysis.- I Elements of Probability Theory.- J Relativistic Point Interactions in One Dimension.- References.