Asch / Joye Mathematical Physics of Quantum Mechanics

Selected and Refereed Lectures from QMath9
2006
ISBN: 978-3-540-34273-1
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 690, 485 Seiten, eBook

Reihe: Lecture Notes in Physics

At the QMath9 meeting, young scientists learn about the state of the art in the mathematical physics of quantum systems. Based on that event, this book offers a selection of outstanding articles written in pedagogical style comprising six sections which cover new techniques and recent results on spectral theory, statistical mechanics, Bose-Einstein condensation, random operators, magnetic Schrödinger operators and much more. For postgraduate students, Mathematical Physics of Quantum Systems serves as a useful introduction to the research literature. For more expert researchers, this book will be a concise and modern source of reference.

Zielgruppe

Research

Autoren/Hrsg.

Asch, Joachim

Joye, Alain

Inhaltsverzeichnis

Quantum Dynamics and Spectral Theory.- Solving the Ten Martini Problem.- Swimming Lessons for Microbots.- Landau-Zener Formulae from Adiabatic Transition Histories.- Scattering Theory of Dynamic Electrical Transport.- The Landauer-Büttiker Formula and Resonant Quantum Transport.- Point Interaction Polygons: An Isoperimetric Problem.- Limit Cycles in Quantum Mechanics.- Cantor Spectrum for Quasi-Periodic Schrödinger Operators.- Quantum Field Theory and Statistical Mechanics.- Adiabatic Theorems and Reversible Isothermal Processes.- Quantum Massless Field in 1+1 Dimensions.- Stability of Multi-Phase Equilibria.- Ordering of Energy Levels in Heisenberg Models and Applications.- Interacting Fermions in 2 Dimensions.- On the Essential Spectrum of the Translation Invariant Nelson Model.- Quantum Kinetics and Bose-Einstein Condensation.- Bose-Einstein Condensation as a Quantum Phase Transition in an Optical Lattice.- Long Time Behaviour to the Schrödinger–Poisson–X? Systems.- Towards the Quantum Brownian Motion.- Bose-Einstein Condensation and Superradiance.- Derivation of the Gross-Pitaevskii Hierarchy.- Towards a Microscopic Derivation of the Phonon Boltzmann Equation.- Disordered Systems and Random Operators.- On the Quantization of Hall Currents in Presence of Disorder.- Equality of the Bulk and Edge Hall Conductances in 2D.- Generic Subsets in Spaces of Measures and Singular Continuous Spectrum.- Low Density Expansion for Lyapunov Exponents.- Poisson Statistics for the Largest Eigenvalues in Random Matrix Ensembles.- Semiclassical Analysis and Quantum Chaos.- Recent Results on Quantum Map Eigenstates.- Level Repulsion and Spectral Type for One-Dimensional Adiabatic Quasi-Periodic Schrödinger Operators.- Low Lying Eigenvalues of Witten Laplacians and Metastability(After Hel.er-Klein-Nier and Helffer-Nier).- The Mathematical Formalism of a Particle in a Magnetic Field.- Fractal Weyl Law for Open Chaotic Maps.- Spectral Shift Function for Magnetic Schrödinger Operators.- Counting String/M Vacua.

Leseproben

Part I Quantum Dynamics and Spectral Theory (p. 3-4)

Different aspects of the solution of a long-standing major problem in mathematical physics are reported in the contributions of Avila, Jitomirskaya and Puig. It had been conjectured for about thirty years by physicists and mathematicians that the problem of electrons confined to a plane under the influence of a periodic potential and a perpendicular magnetic field exhibits fractal spectral properties. Experimental evidence of Hofstadters butterflylike energy spectrum was found about five years ago.

Here the mathematical physicists Avila, Jitomirskaya and Puig report on the proof that the spectrum of a related operator is a Cantor set. Their proofs rely much on recent techniques in classical dynamical systems.We mention that the mathematical model still has fascinating unsolved aspects which are important to physics and especially the quantum hall effect for example the question whether the spectral gaps are open. Building Micron-size robots which move much faster than bacteria is one of the visions of small scale physics. Y. Avron gave an introduction on recent results on the problem of designing an optimal micro-swimmer.

These have been obtained using methods from geometry and linear response theory. V. Betz and S. Teufel report on their progress in the old Landau–Zener problem. For a time dependent two state problem which is asymptotically constant, a detailed approximate solution which takes into account the adiabatic transitions is obtained for all times. It describes both the exponential smallness of the transition probability and the time scale over which it takes place. The theory of transport in mesoscopic systems is addressed in two contributions. Büttiker and Moskalets treat quantum pumping. If the system is driven by several internal parameters oscillating slowly, a direct current may result.

It can be calculated to leading order in terms of stationary scattering matrices. To take account the energy exchange with the environment the full time dependent scattering matrix is developed to next order. A mathematical proof of the formula relating conductance and transmittance has been given by H.D. Cornean, A. Jensen, V. Moldoveanu in the case of an adiabatically switched on external potential. The formula is applied numerically to a model.

Geometry meets physics again in the contribution of P. Exner who presents a conjecture about an interesting isoperimetric problem arising from the spectral analysis of a quantum model with point interactions. S. Glazek discusses examples of renormalization group analysis applied to Schrödinger operators and in particular the occurrence of a limit cycle as a critical attractor instead of a fixed point.