Abstracts, Lecture notes of Quantum Mechanics

Download Abstracts and more Lecture notes Quantum Mechanics in PDF only on Docsity! SimonFest California Institute of Technology March 27 – 31, 2006 Abstracts Critical Phenomena via Random Walk Representations: Selected Classics and Some Recent Results Michael Aizenman Princeton University SimonFest offers an opportunity to reminisce about the contribution of “soft” methods in the study of critical phenomena in Statistical Mechanics— methods which have been warmly embraced by Barry Simon and signifi- cantly advanced in his work on the subject. In addition to a partial review of the results derived at that period, I shall describe also some related re- cent results. The latter includes a spillover of statistical mechanics insights into the analysis of Schrödinger operators, and also the recent proofs by M. Heydenreich and R.W. van der Hofstad, and of the speaker with V. Pap- athanakos, of a long outstanding conjecture concerning a drastic effect of the boundary conditions (periodic versus bulk/free) on the nature of the scaling limits of critical models, in particular above the upper-critical dimensions. Barry and the Quantum Hall Effect: An Argument with the Laughlin Argument Yosi Avron Technion – Israel Institute of Technology I shall review the geometric approach to the integer quantum Hall effect. In particular, I shall survey joint work with Barry Simon and Ruedi Seiler on the interpretation of the Hall conductance as an index. Schur Processes Alexei Borodin California Institute of Technology Measures on partitions with weights given by suitable products of Schur symmetric functions have recently found a variety of applications from Toeplitz determinants to representation theory, stochastic growth processes, 1 2 and random tiling models. The goal of the talk is to give a survey of known results on such measures. A New Approach to Spectral Gap Problems Jean Bourgain Institute for Advanced Study, Princeton Using combinatorial techniques such as the sum-product phenomenon in finite fields, we exhibit new classes of expanders in SL2(p) and SU(2). Vari- ous applications are given, in particular, to the quaquaversal tiling problem. Lyapunov Exponents and Spectral Analysis of Ergodic Schrödinger Operators David Damanik California Institute of Technology The spectral analysis of an ergodic family of one-dimensional Schrödinger operators typically starts out with an investigation of the Lyapunov expo- nent of the associated energy-indexed Schrödinger cocycle over the given ergodic transformation. For example, the absolutely continuous spectrum is given by the essential closure of the set of energies for which the Lya- punov exponent vanishes. We review some general results in this context, particularly Kotani theory, and their application to concrete models. Non-Self-Adjoint Operators and Pseudospectra E B Davies King’s College, London The theory of pseudospectra has grown rapidly since its emergence from within numerical analysis around 1990. We describe some of its applications to the stability theory of differential operators, to WKB analysis and even to orthogonal polynomials. Although currently more a way of looking at non-self-adjoint operators than a list of theorems, its future seems to be assured by the growing number of problems in which the ideas are clearly of relevance. 5 Geometric Scattering Theory: From Particles to Fields Gian Michele Graf ETH, Zürich Scattering theory has been shaped by spectral and geometric approaches. Many among the main results in the last fifteen years or so depended on a combination of both. We shall review some of the contributions by Barry Simon (and others) to these methods, as well as some of their applications, such as to the quantum mechanical N -body problem, or to Rayleigh and Compton scattering. Some open problems will be mentioned, too. Molecular Quantum Mechanics in the Born–Oppenheimer Limit George Hagedorn Virginia Tech Born–Oppenheimer approximations describe molecular quantum mechan- ics in the limit of large nuclear masses. Although these approximations are almost eighty years old and fundamental to theoretical chemistry, their rigor- ous mathematical analysis began only thirty years ago. Most of this analysis has concentrated on validating existing physical theories, but some has led to new insights concerning molecular dynamics. We review the mathematical work in this subject and describe some di- rections in which we hope some future progress might be made. Perturbation Theory and Atomic Resonances Since Schrödinger’s Time Evans M. Harrell Georgia Institute of Technology Quantum theory makes a sharp distinction between bound states and scattering states, the former associated with point spectrum and the lat- ter with continuous spectrum. Resonances associated with quasi-stationary states bridge this distinction, and have posed mathematical challenges since the beginning of the Schrödinger theory. Here the development of mathe- matical aspects of resonances in atomic physics is reviewed, with particular reference to the role of the Stark effect, and perturbations of bound states. 6 Barry Simon’s Contribution to Magnetic and Electric Fields and the Semiclassical Limit Ira W. Herbst University of Virginia In this talk I will review some of Barry Simon’s seminal contributions to the theory of Schrödinger operators with external magnetic and electric fields. I will also talk about a beautiful theorem of Barry’s on the energy splitting in multidimensional double wells in the large coupling limit. Bound State Problems in Quantum Mechanics Dirk Hundertmark University of Illinois at Urbana-Champaign We give a review of semi-classical estimates for bound states and their eigenvalues for Schrödinger operators. Motivated by the classical results, we discuss their recent improvements for single particle Schrödinger operators as well as some applications of these semi-classical bounds to multi-particle systems, in particular, large atoms and the stability of matter. Fermi Golden Rule in Quantum Statistical Mechanics Vojkan Jaksic McGill University The mathematical foundations of time-dependent perturbation theory were laid down by Barry Simon in his seminal 1973 Annals paper. In this talk I will discuss the time-dependent perturbation theory and Fermi Golden Rule in the context of quantum statistical mechanics. The talk will be of a review nature and I will focus on the historical aspects as well as the applications to the problem of return to equilibrium (zeroth law of thermo- dynamics). This talk is based on joint work with C.-A. Pillet. Ergodic Schrödinger Operators: Recent Advances Svetlana Jitomirskaya University of California, Irvine We will review recent progress on the spectral theory of quasiperiodic operators and on the Anderson model. 7 Orthogonal Polynomials: First Minutes Sergey Khrushchev Atilim University, Ankara, Turkey It is standard to refer to Chebychev, Gauss and Jacobi as the creators of Orthogonal Polynomials. In fact, this topic goes back to the very early times of analysis, namely to March of 1655 when Wallis completed his fa- mous book “Arithmetica of Infinitorum.” This book contained a remarkable Section 191, in which Wallis presented his understanding of a solution to the functional equation b(s)b(s + 2) = (s + 1)2 found by Brouncker. Wallis’ presentation was not very clear and posed questions on Brouncker’s proof rather than explaining it. Later in his main paper on Continued Fractions (1739), Euler paid great attention to this result of Brouncker and mentioned that it would be highly desirable to recover Brouncker’s original arguments. In this talk, we present such a recovery and show how this problem is related to orthogonal polynomials. Some Sum Rules Rowan Killip UCLA I will describe some examples of sum rules—simple equations relating the coefficients of operators to their spectral data—and outline their applications in forward/inverse spectral analysis. Imbedded Singular Spectrum for Schrödinger Operators Alexander Kiselev University of Wisconsin at Madison We will review recent results on the imbedded singular spectrum. This will include examples with a dense set of imbedded eigenvalues, and exam- ples where wave operators exist but are not asymptotically complete due to the presence of a singular continuous spectrum. We will also discuss esti- mates on the size of the set where the singular spectrum may be supported, which can be thought of as nonlinear versions of well-known estimates for the Fourier transform.