Mathematical Results In Quantum Physics

The volume collects papers from talks given at QMath11 — Mathematical Results in Quantum Physics, which was held in Hradec Králové, September 2010. These papers bring new and interesting results in quantum mechanics and information, quantum field theory, random systems, quantum chaos, as well as in the physics of social systems. Part of the contribution is dedicated to Ari Laptev on the occasion of his 60th birthday, in recognition of his mathematical results and his service to the community. The QMath conference series has played an important role in mathematical physics for more than two decades, typically attracting many of the best results achieved in the last three-year period, and the meeting in Hradec Králové was no exception. Contents:Relative Entropies and Entanglement Monotones (Nilanjana Datta)Interacting Electrons on the Honeycomb Lattice (Alessandro Giuliani)Convergence Results for Thick Graphs (Olaf Post)Spectral Properties of Wigner Matrices (Benjamin Schlein)Semiclassical Spectral Bounds and Beyond (Timo Weidl)Spectral Problems in Spaces of Constant Curvature (Rafael D Benguria)Localization in Random Displacement Model (Michael Loss & Günther Stolz)Diffusion in Hamiltonian Quantum Systems (Wojciech De Roeck)Quantized Open Chaotic Systems (Stéphen Nonnenmacher)Reliability Issues in the Microscopic Modeling of Pedestrian Movement (Bernhard Steffen, Armin Seyfried & Maik Boltes)and other papers Readership: Graduate students, professionals and researchers in mathematical physics, quantum mechanics and field theory, quantum information, quantum chaos and physics of social systems. Keywords:Quantum Physics;Quantum Mechanics;Quantum Field Theory;Quantum Chaos;Quantum Information;Sociophysics

Quantum mechanics and the theory of operators on Hilbert space have been deeply linked since their beginnings in the early twentieth century. States of a quantum system correspond to certain elements of the configuration space and observables correspond to certain operators on the space. This book is a brief, but self-contained, introduction to the mathematical methods of quantum mechanics, with a view towards applications to Schrodinger operators. Part 1 of the book is a concise introduction to the spectral theory of unbounded operators. Only those topics that will be needed for later applications are covered. The spectral theorem is a central topic in this approach and is introduced at an early stage. Part 2 starts with the free Schrodinger equation and computes the free resolvent and time evolution. Position, momentum, and angular momentum are discussed via algebraic methods. Various mathematical methods are developed, which are then used to compute the spectrum of the hydrogen atom. Further topics include the nondegeneracy of the ground state, spectra of atoms, and scattering theory. This book serves as a self-contained introduction to spectral theory of unbounded operators in Hilbert space with full proofs and minimal prerequisites: Only a solid knowledge of advanced calculus and a one-semester introduction to complex analysis are required. In particular, no functional analysis and no Lebesgue integration theory are assumed. It develops the mathematical tools necessary to prove some key results in nonrelativistic quantum mechanics. Mathematical Methods in Quantum Mechanics is intended for beginning graduate students in both mathematics and physics and provides a solid foundation for reading more advanced books and current research literature. It is well suited for self-study and includes numerous exercises (many with hints).

The 10th Quantum Mathematics International Conference (Qmath10) gave an opportunity to bring together specialists interested in that part of mathematical physics which is in close connection with various aspects of quantum theory. It was also meant to introduce young scientists and new tendencies in the field.This collection of carefully selected papers aims to reflect recent techniques and results on SchrAdinger operators with magnetic fields, random SchrAdinger operators, condensed matter and open systems, pseudo-differential operators and semiclassical analysis, quantum field theory and relativistic quantum mechanics, quantum information, and much more. The book serves as a concise and well-documented tool for the more experimented scientists, as well as a research guide for postgraduate students.

The 10th Quantum Mathematics International Conference (Qmath10) gave an opportunity to bring together specialists interested in that part of mathematical physics which is in close connection with various aspects of quantum theory. It was also meant to introduce young scientists and new tendencies in the field.This collection of carefully selected papers aims to reflect recent techniques and results on Schrödinger operators with magnetic fields, random Schrödinger operators, condensed matter and open systems, pseudo-differential operators and semiclassical analysis, quantum field theory and relativistic quantum mechanics, quantum information, and much more. The book serves as a concise and well-documented tool for the more experimented scientists, as well as a research guide for postgraduate students.

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.

The last decades have demonstrated that quantum mechanics is an inexhaustible source of inspiration for contemporary mathematical physics. Of course, it seems to be hardly surprising if one casts a glance toward the history of the subject; recall the pioneering works of von Neumann, Weyl, Kato and their followers which pushed forward some of the classical mathematical disciplines: functional analysis, differential equations, group theory, etc. On the other hand, the evident powerful feedback changed the face of the "naive" quantum physics. It created a contem porary quantum mechanics, the mathematical problems of which now constitute the backbone of mathematical physics. The mathematical and physical aspects of these problems cannot be separated, even if one may not share the opinion of Hilbert who rigorously denied differences between pure and applied mathemat ics, and the fruitful oscilllation between the two creates a powerful stimulus for development of mathematical physics. The International Conference on Mathematical Results in Quantum Mechan ics, held in Blossin (near Berlin), May 17-21, 1993, was the fifth in the series of meetings started in Dubna (in the former USSR) in 1987, which were dedicated to mathematical problems of quantum mechanics. A primary motivation of any meeting is certainly to facilitate an exchange of ideas, but there also other goals. The first meeting and those that followed (Dubna, 1988; Dubna, 1989; Liblice (in the Czech Republic), 1990) were aimed, in particular, at paving ways to East-West contacts.

The book provides a comprehensive overview on the state of the art of the quantum part of mathematical physics. In particular, it contains contributions to the spectral theory of Schrödinger and random operators, quantum field theory, relativistic quantum mechanics and interacting many-body systems. It also presents an overview on the achievements in mathematical physics since the last conference QMath11 held at Hradec Kralove, Czechia in 2010. Contents:Plenary Talks:A Bound for the Eigenvalue Counting Function for Higher-Order Krein Laplacians on Open Sets (F Gesztesy, M Mitrea, S Sukhtaiev and A Laptev)Trace Formulae for the Asymptotic Density of Eigenvalue Clusters for the Perturbed Landau Hamiltonian (T Lungenstrass and G D Raikov)On Long Range Behaviour of van der Waals Force (I Anapolitanos and I M Sigal)Quantum Spin Correlations and Random Loops (D Ueltschi)Equidistribution Estimates for Eigenfunctions and Eigenvalue Bounds for Random Operators (D Borisov, M Tautenhahn and I Veselić)Invited Section Talks:Vector Quantum Fields (J Dereziński)On the BCS Gap Equation for Superfluid Fermionic Gases (G Bräunlich, C Hainzl and R Seiringer)Improved Hardy Inequality in Twisted Tubes (H Kovařík)Microscopic Foundations of Ohm and Joule's Laws — The Relevance of Thermodynamics (J-B Bru and W de Siqueira Pedra)The Quantum Marginal Problem (C Schilling)Hartree-Fock Dynamics for Weakly Interacting Fermions (N Benedikter, M Porta and B Schlein)Contributed Talks:On the Ground State Energy of the Multipolaron in the Strong Coupling Limit (I Anapolitanos and B Landon)A Variation on Smilansky's Model (D Barseghyan and P Exner)Deriving the Gross-Pitaevskii Equation (N Benedikter)Boundary Triplets Approach for Dirac Operator (A A Boitsev)Bose-Einstein Condensation on Quantum Graphs (J Bolte and J Kerner)Description of Quantum and Classical Dynamics via Feynman Formulae (Ya A Butko)Asymptotic Observables, Propagation Estimates and the Problem of Asymptotic Completeness in Algebraic QFT (W Dybalski)Recent Probabilistic Results on Covariant Schrödinger Operators on Infinite Weighted Graphs (B Güneysu and O Milatovic)Resolvent Expansion for the Discrete One-Dimensional Schrödinger Operator (K Ito and A Jensen)Spectral Asymptotics for a δ′ Interaction Supported by an Infinite Curve (M Jex)Asymptotically Predefined Spectral Gaps for the Neumann Laplacian in Periodic Domains (A Khrabustovskyi)Graph Model for the Stokes Flow (M O Kovaleva and I Yu Popov)Point Contacts and Boundary Triples (V Lotoreichik, H Neidhardt and I Yu Popov)Trace Formulas for Singular and Additive Non-Selfadjoint Perturbations (M M Malamud and H Neidhardt)On δ′-Couplings at Graph Vertices (S S Manko)Stochastic Calculus and Non-Relativistic QED (B Güneysu, O Matte and J S Møller)Estimates for Numbers of Negative Eigenvalues of Laplacian for Y-Type Chain of Weakly Coupled Ball Resonators (A S Melikhova)On Thermodynamical Couplings of Quantum Mechanics and Macroscopic Systems (A Mielke)Almost Sure Purely Singular Continuous Spectrum for Quasicrystal Models (C Seifert)Adiabatic Theorems With and Without Spectral Gap Condition for Non-Semisimple Spectral Values (J Schmid)An Eigenvalue Counting Theorem with Applications to Random Schrödinger Operators (D Schmidt)System of Fermions with Zero-Range Interactions (A Teta) Readership: Graduate students, professionals and researchers in mathematical physics, quantum mechanics and field theory, quantum information, quantum chaos and physics of social systems. Key Features:Collection of state-of-the-art papers in mathematical physicsProminent contributorsShows the actual research topics in mathematical physicsKeywords:Schrödinger Operator;Spectral Theory;Random Operators;Quantum Field Theory;Relativistic Quantum Mechanics;Interacting Many-Body Systems

Presents a comprehensive treatment of quantum mechanics from a mathematics perspective. Including traditional topics, like classical mechanics, mathematical foundations of quantum mechanics, quantization, and the Schrodinger equation, this book gives a mathematical treatment of systems of identical particles with spin.