Order, disorder, and chaos in quantum systems : [proceedings of a conference held at Dubna, USSR, on October 17-21, 1989

The present volume collects the contributions to the conference "Order, disorder and chaos in quantum systems" which was held at Dubna last October. It is the third meeting in the series started three years ago in which we tried to put together mathematical physicists from the member and non-member countries of JINR with their colleagues from soviet universities and institutes using this international centre as a convenient basis. As in the previous cases, new faces, subj ects and ideas appeared but the spirit remained the same, relaxed and inspirative. Among this conference contributions, a majority should be listed in the "orderly" category. Being more specific, this means mostly various aspects of the theory of Schroedinger operators that has been always a core of quantum mechanics. In spite of the fact that it is studied already for several decades, there are still many interesting problems to solve as some of the lectures collected below witness. At the same time, the theory extends to some new areas motivated by physical problems ; let us mention Schroedinger operators in complicated spatial domains appearing in some parts of solid-state physics or various models using the concept of contact interactions. Our world is far from perfect and to keep a perfect order is difficult not only in everyday life but also in most physical systems. Theoreticians are used to take this fact into account introducing stochastic factors into their considerations. Front Matter....Pages I-XIV Front Matter....Pages 1-1 Negative Discrete Spectrum of the Schroedinger Operator with Large Coupling Constant: A Qualitative Discussion....Pages 3-16 Discrete Spectrum in the Gaps of the Continuous one in the Large-Coupling-Constant Limit....Pages 17-25 On the Asymptoptics of Discrete Spectrum for the Schroedinger Operator in Electric and Homogeneous Magnetic Fields....Pages 27-31 Asymptotics of the Discrete Spectrum of Hamiltonians of Quantum Systems with a Homogeneous Magnetic Field....Pages 33-53 Asymptotics of Eigenvalues for Many-Particle Hamiltonians at Symmetry Subspaces....Pages 55-59 Spectral Asymptotics with Highly Accurate Remainder Estimates....Pages 61-63 Bound States and Resonances in Quantum Wires....Pages 65-84 Spectral Properties of the Operators H ψ = − ψ xx + p(x) ψ + v(εx) ψ , p is Periodic....Pages 85-107 Discrete Spectrum for a Periodic Schroedinger Operator Perturbed by a Decreasing Potential....Pages 109-114 On a Complete Description of the Principal Discrete Series of Spectral Invariants of the Hill Operator....Pages 115-117 WKB-Approximations from the Perturbation Theory Viewpoint....Pages 119-123 Moving Potentials and the Completeness of Wave Operators. Part II: Propagating Observables on Scattering States....Pages 125-147 Representation of the Three-Body S-Matrix in Terms of Effective Amplitudes....Pages 149-153 A Three-Body One-Dimensional System with Increasing Interactions: Wave Function Asymptotics....Pages 155-158 Surprises of Quantum Tunneling (Something about the Velocity of the Sub-Barrier Motion)....Pages 159-168 Front Matter....Pages 169-169 Lifshitz-Tails and Non-Lifshitz-Tails for One-Dimensional Random Point Interactions....Pages 171-178 Point Interactions with an Internal Structure as Limits of Nonlocal Separable Potentials....Pages 179-182 Green’s Function for the Aharonov-Bohm Effect with a Non-Abelian Gauge Group....Pages 183-193 A Model of Zero-Width Slits and the Real Diffraction Problem....Pages 195-196 Construction of an Inelastic Scatterer in Nanoelectronics by the Extension-Theory Methods....Pages 197-198 Front Matter....Pages 169-169 On Boundary Theory for Schrödinger Operators and Stochastic Processes....Pages 199-208 Dirac Hamiltonian with Coulomb Potential and Contact Interaction on a Sphere....Pages 209-219 Self-Adjoint Extensions of Schroedinger Operators with Singular Potentials....Pages 221-227 Currents and the Extensions Theory....Pages 229-233 Front Matter....Pages 235-235 Chaotic Quantum Billiards....Pages 237-258 Relevance of the Localization to Quasienergy Statistics in Quantum Chaotic Systems....Pages 259-277 Singular Continuous Quasi-Energy Spectrum in the Kicked Rotator with Separable Perturbation: Onset of Quantum Chaos?....Pages 279-299 Relation between Correlation Functions and Spectrum Statistics in the Region of Quantum Chaos....Pages 301-304 Front Matter....Pages 305-305 Localization Effects in Non-Homogeneous Dielectrics....Pages 307-313 A Self-Adjoint Dilation of the Linear Boltzmann Operator....Pages 315-318 Kato Problem for Functional-Differential Equations and Difference Schrödinger Operators....Pages 319-321 An Existence Theorem for Some Nonlinear Nonlocal Schroedinger Operators and the Soliton-Like Solutions for the Corresponding Dynamic Systems....Pages 323-326 The Sturm-Liouville Problem with a Potential Linear in Spectral Parameter....Pages 327-329 Stochastic Model of Tree Growth....Pages 331-334 Critical Phenomenon in the Stationary Model of Random Growth....Pages 335-338 On a Quantum-Classical Connection, Hidden Symmetries, and a Model of Josephson Junction....Pages 339-342 Open Multiquantum Systems. Method of a Generating Functional....Pages 343-345 Generalized Functions and Their Applications....Pages 347-354 Back Matter....Pages 355-360