This book constitutes the proceedings of the QMath 7 Conference on Mathematical Results in Quantum Mechanics held in Prague, Czech Republic in June, 1998. The volume addresses mathematicians and physicists interested in contemporary quantum physics and associated mathematical questions, presenting new results on Schrödinger and Pauli operators with regular, fractal or random potentials, scattering theory, adiabatic analysis, and interesting new physical systems such as photonic crystals, quantum dots and wires.
Table of ContentsI Plenary talks.- An adiabatic theorem without a gap condition.- Two-dimensional periodic Pauli operator. The effective masses at the lower edge of the spectrum.- Spectral problems in the theory of photonic crystals.- Optimal eigenvalues for some Laplacians and Schrödinger operators depending on curvature.- The spectral shift operator.- On the scattering operator for the Schrödinger equation with a time-dependent potential.- H_2-construction and some applications.- Scattering with time periodic potentials and cyclic states.- Some geometry related to decay properties of the resolvent of a class of symmetric operators.- On some asymptotic formulas in the strong localization regime of the theory of disordered systems.- Spectral measures and category.- Quantum dots. A survey of rigorous results.- II Session talks.- A simple model of concentrated nonlinearity.- Anomalous electron trapping by magnetic flux tubes and electric current vortices.- On the absolutely continuous energy distribution of a quantum mechanical system in a bounded domain.- Some aspects of generalized contact interaction in one-dimensional quantum mechanics.- Traces and trace norms for semigroup differences...- About a resolvent formula.- The determinant anomaly in low-dimensional quantum systems.- Linear Boltzmann equation as the weak coupling limit of the random Schrödinger equation.- Coexistence of different spectral types for almost periodic Schrödinger equations in dimension one.- Dynamical localization for random Schrödinger operators and an application to the almost Mathieu operator.- On fractal structure of the spectrum for periodic point perturbations of the Schrödinger operator with a uniform magnetic field.- A Weyl-Berry formula for the scattering operator associated to self-similar potentials on the line.- Localization and Lifshitz tails for random quantum waveguides.- Birman-Schwinger analysis for bound states in a pair of parallel quantum waveguides with a semitransparent boundary.- On the absolute continuity of spectra of periodic elliptic operators.- Hardy inequalities for magnetic Dirichlet forms.- Adiabatic curvature, chaos and the deformation of Riemann surfaces.- Operator interpretation of resonances arising in spectral problems for 2 x 2 matrix Hamiltonians.- On the operator-norm convergence of the Trotter-Kato product formula.- A particular case of the inverse problem for the Sturm-Liouville equation with parameter dependent potential.- One-dimensional Schrödinger operators with decaying potentials.- Stability of limiting absorption under singular perturbations.- Existence of averaging integrals for self-adjoint operators.- Monotonicity versus non-monotonicity in random operators.- A model in perturbation theory.- Band gap of the spectrum in periodically curved quantum waveguides.- A list of other talks.- A list of participants.