Ebook: Mathematical Results in Quantum Mechanics: QMath7 Conference, Prague, June 22–26, 1998

At the age of almost three quarters of a century, quantum mechanics is by all accounts a mature theory. There were times when it seemed that it had borne its best fruit already and would give way to investigation of deeper levels of matter. Today this sounds like rash thinking. Modern experimental techniques have led to discoveries of numerous new quantum effects in solid state, optics and elsewhere. Quantum mechanics is thus gradually becoming a basis for many branches of applied physics, in this way entering our everyday life. While the dynamic laws of quantum mechanics are well known, a proper theoretical understanding requires methods which would allow us to de­ rive the abundance of observed quantum effects from the first principles. In many cases the rich structure hidden in the Schr6dinger equation can be revealed only using sophisticated tools. This constitutes a motivation to investigate rigorous methods which yield mathematically well-founded properties of quantum systems.

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This book contains the proceedings of the QMath 7 Conference on Mathematical Results in Quantum Mechanics held in Prague, Czech Republic, from June 22 to 26, 1998. The purpose is to draw attention to recent developments in quantum mechanics stemming from its numerous applications, and to related mathematical problems and techniques.
This volume is addressed to the broad audience of mathematicians and physicists interested in contemporary quantum physics and associated mathematical questions. The reader will find new results on Schr?dinger and Pauli operators with regular, fractal or random potentials, scattering theory, adiabatic analysis, as well as on interesting new physical systems such as photonic crystals, quantum dots and wires

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This book contains the proceedings of the QMath 7 Conference on Mathematical Results in Quantum Mechanics held in Prague, Czech Republic, from June 22 to 26, 1998. The purpose is to draw attention to recent developments in quantum mechanics stemming from its numerous applications, and to related mathematical problems and techniques.
This volume is addressed to the broad audience of mathematicians and physicists interested in contemporary quantum physics and associated mathematical questions. The reader will find new results on Schr?dinger and Pauli operators with regular, fractal or random potentials, scattering theory, adiabatic analysis, as well as on interesting new physical systems such as photonic crystals, quantum dots and wires

Content:
Front Matter....Pages i-x
Front Matter....Pages 1-1
An Adiabatic Theorem without a Gap Condition....Pages 3-12
Two-Dimensional Periodic Pauli Operator. The Effective Masses at the Lower Edge of the Spectrum....Pages 13-31
Spectral Problems in the Theory of Photonic Crystals....Pages 33-46
Optimal Eigenvalues for Some Laplacians and Schr?dinger Operators Depending on Curvature....Pages 47-58
The Spectral Shift Operator....Pages 59-90
On the scattering operator for the Schr?dinger equation with a time-dependent potential....Pages 91-97
Scattering with time periodic potentials and cyclic states....Pages 99-105
Some geometry related to decay properties of the resolvent of a class of symmetric operators....Pages 107-115
On Some Asymptotic Formulas in the Strong Localization Regime of the Theory of Disordered Systems....Pages 117-128
Spectral Measures and Category....Pages 129-148
Quantum dots....Pages 149-159
Front Matter....Pages 161-180
A Simple Model of Concentrated Nonlinearity....Pages 181-181
Anomalous electron trapping by magnetic flux tubes and electric current vortices....Pages 183-189
On the absolutely continuous energy distribution of a quantum mechanical system in a bounded domain....Pages 191-196
Some Aspects of Generalized Contact Interaction in One-Dimensional Quantum Mechanics....Pages 197-202
Traces and trace norms for semigroup differences....Pages 203-208
About a resolvent formula....Pages 209-219
The determinant anomaly in low-dimensional quantum systems....Pages 221-226
Linear Boltzmann Equation as the Weak Coupling Limit of the Random Schr?dinger Equation....Pages 227-232
Front Matter....Pages 233-242
Coexistence of different spectral types for almost periodic Schr?dinger equations in dimension one....Pages 181-181
Dynamical Localization for Random Schr?dinger Operators and an Application to the Almost Mathieu Operator....Pages 243-251
On fractal structure of the spectrum for periodic point perturbations of the Schr?dinger operator with a uniform magnetic field....Pages 253-257
A Weyl-Berry formula for the scattering operator associated to self-similar potentials on the line....Pages 259-265
Localization and Lifshitz tails for Random Quantum Waveguides....Pages 267-274
Birman-Schwinger Analysis for Bound States in a Pair of Parallel Quantum Waveguides with a Semitransparent Boundary....Pages 275-279
On absolute continuity of spectra of periodic elliptic operators....Pages 281-289
Hardy inequalities for magnetic Dirichlet forms....Pages 291-297
Adiabatic curvature, chaos and the deformations of Riemann Surfaces....Pages 299-305
Operator Interpretation of Resonances Arising in Spectral Problems for 2 x 2 Matrix Hamiltonians....Pages 307-314
On the operator-norm convergence of the Trotter-Kato product formula....Pages 315-322
A Particular Case of The Inverse Problem for The Sturm-Liouville Equation with Parameter-Dependent Potential....Pages 323-334
One-dimensional Schr?dinger operators with decaying potentials....Pages 335-341
Stability of Limiting Absorption Principle under Singular Perturbations....Pages 343-349
Existence of Averaging Integrals for Self-Adjoint Operators....Pages 351-357
Monotonicity versus Non-Monotonicity in Random Operators....Pages 359-364
A model in perturbation theory....Pages 365-371
Band Gap of the Spectrum in Periodically Curved Quantum Waveguides....Pages 373-378
Back Matter....Pages 379-384
....Pages 385-398