Solvable Models in Quantum Mechanics

I The One-Center Point Interaction.- I.1 The One-Center Point Interaction in Three Dimensions.- I.1.1 Basic Properties.- I.1.2 Approximations by Means of Local as well as Nonlocal Scaled Short-Range Interactions.- I.1.3 Convergence of Eigenvalues and Resonances.- I.1.4 Stationary Scattering Theory.- Notes.- I.2 Coulomb Plus One-Center Point Interaction in Three Dimensions.- I.2.1 Basic Properties.- I.2.2 Approximations by Means of Scaled Coulomb-Type Interactions.- I.2.3 Stationary Scattering Theory.- Notes.- I.3 The One-Center ?-Interaction in One Dimension.- I.3.1 Basic Properties.- I.3.2 Approximations by Means of Local Scaled Short-Range Interactions.- I.3.3 Convergence of Eigenvalues and Resonances.- I.3.4 Stationary Scattering Theory.- Notes.- I.4 The One-Center ??-Interaction in One Dimension.- Notes.- I.5 The One-Center Point Interaction in Two Dimensions.- Notes.- II Point Interactions with a Finite Number of Centers.- II.1 Finitely Many Point Interactions in Three Dimensions.- II.1.1 Basic Properties.- II.1.2 Approximations by Means of Local Scaled Short-Range Interactions.- II.1.3 Convergence of Eigenvalues and Resonances.- II.1.4 Multiple Well Problems.- II.1.5 Stationary Scattering Theory.- Notes.- II.2 Finitely Many ?-Interactions in One Dimension.- II.2.1 Basic Properties.- II.2.2 Approximations by Means of Local Scaled Short-Range Interactions.- II.2.3 Convergence of Eigenvalues and Resonances.- II.2.4 Stationary Scattering Theory.- Notes.- II.3 Finitely Many ??-Interactions in One Dimension.- Notes.- II.4 Finitely Many Point Interactions in Two Dimensions.- Notes.- III Point Interactions with Infinitely Many Centers.- III.1 Infinitely Many Point Interactions in Three Dimensions.- III.1.1 Basic Properties.- III.1.2 Approximations by Means of Local Scaled Short-Range Interactions.- III.1.3 Periodic Point Interactions.- III.1.4 Crystals.- III.1.5 Straight Polymers.- III.1.6 Monomolecular Layers.- III.1.7 Bragg Scattering.- III.1.8 Fermi Surfaces.- III.1.9 Crystals with Defects and Impurities.- Notes.- III.2 Infinitely Many ?-Interactions in One Dimension.- III.2.1 Basic Properties.- III.2.2 Approximations by Means of Local Scaled Short-Range Interactions.- III.2.3 Periodic ?-Interactions.- III.2.4 Half-Crystals.- III.2.5 Quasi-periodic ?-Interactions.- III.2.6 Crystals with Defects and Impurity Scattering.- Notes.- III.3 Infinitely Many ??-Interactions in One Dimension.- Notes.- III.4 Infinitely Many Point Interactions in Two Dimensions.- Notes.- III.5 Random Hamiltonians with Point Interactions.- III.5.1 Preliminaries.- III.5.2 Random Point Interactions in Three Dimensions.- III.5.3 Random Point Interactions in One Dimension.- Notes.- 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.