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.
Table of Contents1. Schroedinger operators : spectra, scattering and semiclassical behaviour.- Negative discrete spectrum of the Schroedinger operator with large coupling constant : qualitative discussion.- Discrete spectrum in the gaps of the continuous one in the large-coupling-constant limit.- On the asymptotics of discrete spectrum for the Schroedinger operator in electric and homogeneous magnetic fields.- Asymptotics of the discrete spectrum of Hamiltonians of quantum systems with a homogeneous magnetic field.- Asymptotics of eigenvalues for many-particle Hamiltonians at symmetry subspaces.- Spectral asymptotics with highly accurate remainder estimates.- Bound states and resonances in quantum wires.- Spectral properties of the operators H? = ??XX + p(x)? + v(?x), p is periodic.- Discrete spectrum for a periodic Schroedinger operator perturbed by a decreasing potential.- On a complete description of the principal discrete series of spectral invariants of the Hill operator.- WKB-approximations from the perturbation theory viewpoint.- Moving potentials and the completeness of wave operators. Part II : Propagating observables on scattering states.- Representation of the three-body S-matrix in terms of effective amplitudes.- A three-body one-dimensional system with increasing interaction : wave function asymptotics.- Surprises of quantum tunneling (something about the velocity of sub-barrier motion).- 2. Point and contact interactions, self-adjoint extensions.- Lifshitz tails and non-Lifshitz tails for one-dimensional random point interactions.- Point interactions with an internal structure as limits of nonlocal point interactions.- Green’s function for the Aharonov-Bohm effect with a non-Abelian gauge group.- A model of zero-width slits and the real diffraction problem.- Construction of an inelastic scatterer in nanoelectronics by the extension-theory methods.- On boundary theory for Schroedinger operators and stochastic processes.- Dirac Hamiltonian with Coulomb potential and contact intraction on a sphere.- Self-adjoint extensions of Schroedinger operators with singular potentials.- Currents and the extension theory.- 3. Quantum chaos.- Chaotic quantum billiards.- Relevance of the localization to quasienergy statistics in quantum chaotic systems.- Singular continuous quasi-energy spectrum in the kicked rotator with separable perturbation : onset of quantum chaos ?.- Relation between correlation functions and spectrum statistics in the region of quantum chaos.- 4. Other topics.- Localization effects in non-homogeneous dielectrics.- A self-adjoint dilation of the linear Boltzmann operator.- Kato problem for functional-differential equations and difference Schroedinger operators.- An existence theorem for some nonlinear nonlocal Schroedinger operators and the solitonlike dynamics for the corresponding dynamical systems.- The Sturm-Liouville problem with a potential linear in spectral parameter.- Stochastic model of tree growth.- Critical phenomenon in the stationary model of random growth.- On a quantum-classical connection, hidden symmetries and a model of Josephson junction.- Open multiquantum systems. Method of a generating functional.- Generalized functions and their applications.- List of unpublished contributions.- List of participants.