Lieb Robinson Bounds For Multi Commutators And Applications To Response Theory
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Lieb-Robinson Bounds for Multi-Commutators and Applications to Response Theory
Lieb-Robinson bounds for multi-commutators are effective mathematical tools to handle analytic aspects of infinite volume dynamics of non-relativistic quantum particles with short-range, possibly time-dependent interactions.In particular, the existence of fundamental solutions is shown for those (non-autonomous) C*-dynamical systems for which the usual conditions found in standard theories of (parabolic or hyperbolic) non-autonomous evolution equations are not given. In mathematical physics, bounds on multi-commutators of an order higher than two can be used to study linear and non-linear responses of interacting particles to external perturbations. These bounds are derived for lattice fermions, in view of applications to microscopic quantum theory of electrical conduction discussed in this book. All results also apply to quantum spin systems, with obvious modifications. In order to make the results accessible to a wide audience, in particular to students in mathematics with little Physics background, basics of Quantum Mechanics are presented, keeping in mind its algebraic formulation. The C*-algebraic setting for lattice fermions, as well as the celebrated Lieb-Robinson bounds for commutators, are explained in detail, for completeness.
Mathematical and Computational Modelling Across the Scales
Many physical and engineering systems deal with micro-, meso-, macro-, and multi-scale phenomena. The accurate description and the reliable simulation of such phenomena entail major challenges from the point of view of both mathematical modelling and computational engineering. This book covers a selection of challenges related to "Mathematical and Computational Modelling Across the Scales”, stemming from the lecture notes of the XX edition of the Jacques-Louis Lions Spanish-French School in Numerical Simulations in Physics & Engineering. The thematic focus is broad, encompassing mathematical models of complex physical problems, theoretical results on their derivation, and development of numerical methods for their efficient simulation. The contributions of the book include: uncertainty quantification for phenomena at different scales such as epidemic dynamics, medical imaging, and geophysical exploration; structural health monitoring integrating small-scale sensor data in large-scale computational models; frontier numerical methods for the simulation of geophysical and heliophysical dynamics accounting for multi-scale, heterogeneous media; multi-physics, multi-scale models for the mechanobiology of atheroma plaques formation; locomotion models for swimming at the micro-scale; mathematical foundations of quantum mechanics phenomena at the micro-scale. The book is addressed to scientists and engineers, from both academia and industry, interested in the mathematical modelling and numerical simulation of a variety of complex systems in physics and engineering characterised by multiple scales.
Mathematical Problems in Quantum Physics
Author: Federico Bonetto
language: en
Publisher: American Mathematical Soc.
Release Date: 2018-10-24
This volume contains the proceedings of the QMATH13: Mathematical Results in Quantum Physics conference, held from October 8–11, 2016, at the Georgia Institute of Technology, Atlanta, Georgia. In recent years, a number of new frontiers have opened in mathematical physics, such as many-body localization and Schrödinger operators on graphs. There has been progress in developing mathematical techniques as well, notably in renormalization group methods and the use of Lieb–Robinson bounds in various quantum models. The aim of this volume is to provide an overview of some of these developments. Topics include random Schrödinger operators, many-body fermionic systems, atomic systems, effective equations, and applications to quantum field theory. A number of articles are devoted to the very active area of Schrödinger operators on graphs and general spectral theory of Schrödinger operators. Some of the articles are expository and can be read by an advanced graduate student.