MenuMathematical Physics and Harmonic Analysis Seminar
Spring 2009
| Date: | January 23, 2009 |
| Time: | 1:50pm |
| Location: | BLOC 627 |
| Speaker: | Alfonso Montes-Rodriguez, University de Seville, Spain |
| Title: | Uncertainty Principle and Uniqueness Sets for the Klein-Gordon Equation |
| Date: | January 30, 2009 |
| Time: | 1:50pm |
| Location: | BLOC 627 |
| Speaker: | Stephen Fulling, Texas A&M University |
| Title: | Two Apparent Paradoxes of Concrete Spectral Theory |
| Abstract: | Two recent papers in the physics literature claim results that prima facie seem to be in conflict with the well established theory of self-adjoint extensions and eigenfunction expansions for ODEs. In one case an obviously limit-point problem is treated as if it were limit-circle: an "alternative" basis of eigenfunctions is presented, with numerical verification. The other paper finds a "continuous spectrum of normalizable eigenfunctions", which probably indicates that a boundary condition has been left out -- i.e., a limit-circle problem is being treated as limit-point. Despite some mathematical naivete, these papers do not exhibit glaring mistakes, and some of their conclusions, properly interpreted, may be valuable insights that would not be noticed by a mathematician following the traditional approach. I report progress in resolving these two situations. Input from experts in inverse problems and hyperbolic PDE would help me to finish the job. |
| Date: | February 13, 2009 |
| Time: | 1:50pm |
| Location: | BLOC 627 |
| Speaker: | Peter Kuchment, Texas A&M University |
| Title: | Parseval frames of exponentially decaying Wannier functions |
| Abstract: | When studying linear constant coefficient PDEs, or more generally
equations invariant with respect to shifts, the Fourier transform
is known to be very useful, as well as two basic sets of functions:
plane waves and delta-functions, which are interchanged by the Fourier
transform. In case of equations invariant with respect to a lattice in Rn only (e.g., the solid state physics' Schroedinger equation with periodic potential or photonic crystal theory's Maxwell operator in a periodic dielectric medium), the Fourier transform and the corresponding function sets are not useful anymore. However, analogs of these are known: Floquet-Bloch transform replaces the Fourier one, Bloch functions play the role of plane waves, and the so called Wannier functions are analogs of delta functions. All of these are heavily used in physics and mathematics. The problem is with the localization of Wannier functions: one wants them to decay as fast as possible (exponentially is prefered); after all, they are analogs of delta functions! And here one hits a snag: it has been known since 1980s (Thoules) that existence of an orthonormal basis of exponentially (in fact, even much slower) decaying Wannier functions faces a topological obstruction in the form of possible non-triviality of a vector bundle on a torus. Efforts are still being spent on finding cases where the bundle is trivial, although generically (e.g., in magnetic fields) it is not. In this talk I will present a recent result showing that if one relaxes the orthonormal basis condition and asks only for a Parseval (analog of orhtonormality) frame of such functions, it always exists. The bundle tells one how redundant the frame should be (if the bundle is trivial, the frame is a basis). The techniques in proving this result come from spectral theory, several complex variables and some elementary algebraic topology. No knowledge of these topics will be assumed, though :-) |
| Date: | March 6, 2009 |
| Time: | 1:50pm |
| Location: | BLOC 627 |
| Speaker: | E. Abakumov, Univerity of Marne-la-Vallee |
| Title: | Approximation by translates in weighted spaces |
| Abstract: | We discuss some problems on approximation by discrete translates of a given function in weighted function spaces. |
| Date: | March 13, 2009 |
| Time: | 3:00pm |
| Location: | BLOC 627 |
| Speaker: | Dmitry Panchenko, Texas A&M University |
| Title: | The Ghirlanda-Guerra identities and ultrametricity in the Sherrington-Kirkpatrick model |
| Abstract: | The Parisi theory of the SK model completely describes the geometry of the Gibbs sample in a sense that it predicts the limiting joint distribution of all scalar products, or overlaps, between i.i.d. replicas. One of the main properties of this distribution is the ultrametricity which means that the Gibbs measure approximately concentrates on the ultrametric subset of all configurations; another property is the Ghirlanda-Guerra distributional identities. It is well known that these two properties completely determine the distribution and, probably for this reason, they were always considered complementary. We show that if an overlap takes finitely many values then the Ghirlanda-Guerra identities actually imply ultrametricity. |
| Date: | March 27, 2009 |
| Time: | 1:50pm |
| Location: | BLOC 627 |
| Speaker: | Misko Mitkovski, Texas A&M University |
| Title: | Polya sets, gap theorems and Toeplitz kernels. |
| Abstract: | A separated sequence of real numbers is called a Polya sequence if any entire function of exponential type zero that is bounded on that sequence is a constant. We give a description of all such sequences thus solving a problem of Polya and Levinson. Furthermore, we show that this problem is equivalent to a version of a Beurling gap problem. The key step is to give a description of both problems in terms of kernels of certain Toeplitz operators. Finally, we use Toeplitz kernels versions of Beurling-Malliavin theorems given recently by Makarov and Poltoratski to obtain the complete metric description. |
| Date: | April 3, 2009 |
| Time: | 3:00pm |
| Location: | BLOC 627 |
| Speaker: | Joel Zinn, Texas A&M University |
| Title: | On the Gaussian Correlation Inequality |
| Date: | April 8, 2009 |
| Time: | 4:00pm |
| Location: | BLOC 627 |
| Speaker: | O. Post, Humbolt University, Berlin |
| Title: | Quantum graph approximations of thin branching structures |
| Abstract: | Many physical systems have branching structure of thin transversal diameter. One can name for instance quantum wire circuits, thin branching waveguides, or carbon nano-structures. In applications, such systems are often approximated by the underlying one-dimensional graph structure, a so-called "quantum graph". In this way, many properties of the system like conductance can be calculated easier (sometimes even explicitly). After briefly explaining the notion of a quantum graph, we show that the system with thin transversal diameter converges to a quantum graph. We also identify which vertex couplings (which influence the current through the graph) can be obtained by appropriate engineering of the branching structure. |
| Date: | April 17, 2009 |
| Time: | 1:50pm |
| Location: | BLOC 627 |
| Speaker: | D. Damanik, Rice University |
| Title: | Pseudorandom Potentials: Open Problems and Some Recent Results |
| Abstract: | A bounded infinite sequence of real numbers is called a pseudorandom potential if the Schrodinger operator with this sequence as a potential has spectral properties akin to those of a typical Schr"odinger operator with a random sequence as a potential. We present several sequences that are conjectured to be pseudorandom potentials based on heuristics and numerics and then explain some related rigorous results. |
| Date: | April 24, 2009 |
| Time: | 1:50pm |
| Location: | BLOC 627 |
| Speaker: | R. Band, Weizmann Institute of Science |
| Title: | Counting nodal domains on quantum graphs |
| Abstract: | The investigation of nodal domains on manifolds has began already in the 19th century by the pioneering work of Chladni on the nodal structures of vibrating plates. Counting nodal domains started with Sturm's oscillation theorem which states that a vibrating string is divided into exactly n nodal intervals by the zeros of its nth vibrational mode. A quantum graph can be thought of as a structure of strings which are attached to each other. For a given quantum graph the nodal count sequence is the sequence of numbers of nodal intervals of its vibrational modes ordered by their frequencies. Many recent works treated the bounds and the statistics of the nodal sequences both for graphs and for manifolds. Nevertheless, an exact formula for the nodal sequence is still not available. We show the existence of a formula for a specific quantum graph and offer an approach which might yield nodal count formulae for quantum graphs. Such a formula would help in answering the inverse question regarding the geometrical information that is stored in the nodal sequence. This is a joint work with Gregory Berkolaiko and Uzy Smilansky. |
| Date: | May 1, 2009 |
| Time: | 1:50pm |
| Location: | BLOC 627 |
| Speaker: | H. Krueger, Rice University |
| Title: | The potential V(n) = f(nρ (mod 1)) |
| Abstract: | Consider the potential V(n) = f(nρ (mod 1)) for ρ > 0 not an integer, and the associated Schrodinger operator H. I will study the spectrum and Lyapunov exponent of this operator, in particular deriving explicit formulas for it in the case 0 < ρ < 1. |
| Date: | May 6, 2009 |
| Time: | 4:15pm |
| Location: | BLOC 627 |
| Speaker: | Y. Lyubarskii, NTNU, Norway |
| Title: | Direct and Inverse Problem of Multichannel scattering |
| Abstract: | We consider direct and inverse scattering problems for a system of particles interacting pairwise with each other and perhaps with an external field. It is assumed that the system consists of a finite set of channels, i.e., semi-infinite chains of particles, attached to an "innerpart" - a finite system of interacting particles, and that the system performs small oscillations around stable equilibrium. We show that having the scattering data one can reconstruct the characteristics of the system along the channels and also find conditions which allow to reconstruct the characteristics of the inner part of the system.
This is a joint work with Vladimir Marchenko (Institute for Low Temperatures, Ukraine) |
| Date: | May 14, 2009 |
| Time: | 1:00pm |
| Location: | BLOC 627 |
| Speaker: | V. Gurarii, Swinburne University of Technology |
| Title: | Divergent expansions: Error bounds, Duality and Stokes phenomenon |
| Abstract: | We consider classes of functions uniquely determined by coefficients of their divergent expansions. By approximating a function in such a class by partial sums of its expansion we study how accuracy changes when we move within a region of the complex plane. This enables us to discover some features of Stokes phenomenon which are not covered in the literature and to propose a theory of divergent expansions, which includes Stokes phenomenon as its essential part. This in turn allows us to formulate necessary and sufficient conditions for divergent expansions to encounter Stokes' phenomenon. |
