MenuMathematical Physics and Harmonic Analysis Seminar
Spring 2020
| Date: | February 6, 2020 |
| Time: | 2:00pm |
| Location: | BLOC 628 |
| Speaker: | Darren Ong, XIAMEN UNIVERSITY MALAYSIA |
| Title: | Restrictions on the existence of a canonical system flow hierarchy |
| Abstract: | The KdV hierarchy is a family of evolutions on a Schr\"odinger operator that preserves its spectrum. Canonical systems are a generalization of Schr\"odinger operators, that nevertheless share many features with Schr\"odinger operators. Since this is a very natural generalization, one would expect that it would also be straightforward to build a hierarchy of isospectral evolutions on canonical systems analogous to the KdV hierarchy. Surprisingly, we show that there are many obstructions to constructing a hierarchy of flows on canonical systems that obeys the standard assumptions of the KdV hierarchy. This is joint work with Injo Hur. |
| Date: | February 11, 2020 |
| Time: | 2:00pm |
| Location: | BLOC 628 |
| Speaker: | Alan Haynes, University of Houston |
| Title: | Diophantine approximation and diffraction from quasicrystals |
| Abstract: | In this talk, we will explain how diffraction patterns observed from cut and project sets (models for physical materials called quasicrystals) are determined by Diophantine approximation properties of the underlying constructions. The classical approach to calculating diffraction patterns seen from these objects assumes an infinite model, and for this reason, it is not the most practical, from an experimental point of view. Our approach (joint work with Michael Baake) quantifies precisely how much the diffraction patterns observed from finite patterns in cut and project sets deviate from the infinite models. Our methods are explicit and geared towards numerical computation, and they demonstrate the importance of Diophantine approximation to accurately determining complex phases and amplitudes of these diffraction images. |
| Date: | February 28, 2020 |
| Time: | 11:30am |
| Location: | BLOC 628 |
| Speaker: | Ram Band, Technion - Israel Institute of Technology |
| Title: | Quotients of finite-dimensional operators by symmetry representations (UNUSUAL TIME) |
| Abstract: | A finite dimensional operator which commutes with some symmetry group, admits quotient operators. Such a quotient operator is determined by the group action and by picking a certain representation of this group. We present a computationally simple construction to obtain quotients that reflect the structure of the original operators. These quotient operators allow us to generalize previous isospectral constructions of discrete graphs, as well as to provide tools for spectral analysis of finite dimensional operators. This talk is based on a joint work with Gregory Berkolaiko, Christopher H. Joyner and Wen Liu. |
| Date: | March 27, 2020 |
| Time: | 1:50pm |
| Location: | BLOC 628 |
| Speaker: | Stephen Shipman, LSU |
| Title: | CANCELLED |
| Date: | April 3, 2020 |
| Time: | 1:50pm |
| Location: | BLOC 628 |
| Speaker: | Jake Fillman, Texas State University |
| Title: | CANCELLED |
| Date: | April 10, 2020 |
| Time: | 1:50pm |
| Location: | BLOC 628 |
| Speaker: | Braxton Osting, University of Utah |
| Title: | CANCELLED |
| Date: | April 10, 2020 |
| Time: | 1:50pm |
| Location: | Zoom seminar |
| Speaker: | Gregory Berkolaiko, Texas A&M University |
| Title: | Zoom Seminar: Global extrema of dispersion relation of tight-binding models |
| Abstract: | Tight-binding approximation is frequently used in physics to analyze wave propagation through periodic medium. Its Floquet–Bloch transform is a compact graph with a parameter-dependent operator defined on it. The graph of the eigenvalues as functions of parameters is called the dispersion relation. Extrema (minima and maxima) of the n-th eigenvalue give rise to band edges: endpoints of intervals supporting continuous spectrum and therefore allowing wave propagation. Locating the extrema can be difficult in general; there are examples where extrema occur away from the set of parameters with special symmetry. In this talk we will show that a large family of tight-binding models have a curious property: there is a local condition akin to the second derivative test that detects if a critical point is a global (sic!) extremum. Under some additional assumptions (time-reversal and dimension 3 or less), we show that any local extremum of a given sheet of the dispersion relation is in fact the global extremum. Based on a joint project with Yaiza Canzani, Graham Cox, Jeremy Marzuola. |
| Date: | April 17, 2020 |
| Time: | 1:50pm |
| Location: | Zoom seminar |
| Speaker: | Jake Fillman, Texas State University |
| Title: | Zoom seminar: Spectra of Fibonacci Hamiltonians |
| Abstract: | The Fibonacci sequence is a prominent model of a 1D quasicrystal. We will talk about some properties of continuum Schr\"odinger operators with potentials that are determined by the Fibonacci sequence. We show that the spectrum is an (unbounded) Cantor set of zero Lebesgue measure and that the local Hausdorff dimension of the spectrum tends to one in the regimes of high energy and small coupling. We also show that multidimensional Schr\"odinger operators patterned on the Fibonacci sequence can exhibit the coexistence of two phenomena: (1) Cantor structure near the bottom of the spectrum and (2) an absence of gaps in the spectrum at high energies. To prove (2), we develop an "abstract" Bethe--Sommerfeld criterion for sums of extended Cantor sets, which may be of independent interest. [Based on joint projects with David Damanik, Anton Gorodetski, and May Mei] |
| Date: | April 24, 2020 |
| Time: | 1:50pm |
| Location: | BLOC 628 |
| Speaker: | Naoki Saito, UC Davis |
| Title: | CANCELLED |
| Date: | April 24, 2020 |
| Time: | 1:50pm |
| Location: | Zoom seminar |
| Speaker: | Stephen Shipman, LSU |
| Title: | Zoom Seminar: Reducible and irreducible Fermi surfaces for periodic operators |
| Abstract: | I will discuss new theorems concerning reducibility of the Fermi surface for periodic Schrödinger operators. (1) Irreducibility for a class of planar discrete graph operators; (2) Reducibility of multilayer graphs due to compatible asymmetries of the connecting edges; (3) Reducibility of multilayer graphs due to separability or bipartiteness of the layers. Parts of this work are in collaboration with Wei Li, Lee Fisher, Karl-Michael Schmidt, Ian Wood, and Malcolm Brown. |
| Date: | May 1, 2020 |
| Time: | 1:50pm |
| Location: | Zoom seminar |
| Speaker: | Milivoje Lukic, Rice University |
| Title: | Zoom Seminar: Stahl--Totik regularity for continuum Schr\"odinger operators |
| Abstract: | This talk describes joint work with Benjamin Eichinger: a theory of regularity for one-dimensional continuum Schr\"odinger operators, based on the Martin compactification of the complement of the essential spectrum. For a half-line Schr\"odinger operator $-\partial_x^2+V$ with a bounded potential $V$, it was previously known that the spectrum can have zero Lebesgue measure and even zero Hausdorff dimension; however, we obtain universal thickness statements in the language of potential theory. Namely, we prove that the essential spectrum is not polar, it obeys the Akhiezer--Levin condition, and moreover, the Martin function at $\infty$ obeys the two-term asymptotic expansion $\sqrt{-z} + \frac{a}{2\sqrt{-z}} + o(\frac 1{\sqrt{-z}})$ as $z \to -\infty$. The constant $a$ in its asymptotic expansion plays the role of a renormalized Robin constant suited for Schr\"odinger operators and enters a universal inequality $a \le \liminf_{x\to\infty} \frac 1x \int_0^x V(t) dt$. This leads to a notion of regularity, with connections to the exponential growth rate of Dirichlet solutions and the zero counting measures for finite restrictions of the operator. We also present applications to decaying and ergodic potentials. |
| Date: | May 8, 2020 |
| Time: | 1:50pm |
| Location: | Zoom seminar |
| Speaker: | Wei Li, LSU |
| Title: | Zoom Seminar: Embedded eigenvalues of the Neumann Poincaré operator |
| Abstract: | The Neumann-Poincaré (NP) operator arises in boundary value problems, and plays an important role in material design, signal amplification, particle detection, etc. The spectrum of the NP operator on domains with corners was studied by Carleman before tools for rigorous discussion were created, and received a lot of attention in the past ten years. In this talk, I will present our discovery and verification of eigenvalues embedded in the continuous spectrum of this operator. The main ideas are decoupling of spaces by symmetry and construction of approximate eigenvalues. This is based on two works with Stephen Shipman and Karl-Mikael Perfekt. |
| Date: | May 15, 2020 |
| Time: | 1:50pm |
| Location: | Zoom seminar |
| Speaker: | Ian Jauslin, Princeton University |
| Title: | Zoom seminar:A simple equation to study interacting Bose gasses |
| Abstract: | In this talk, I will discuss a partial differential equation introduced by Lieb in 1963 in the context of studying interacting Bose gasses. I will first discuss how this equation can be used to accurately compute physically relevant quantities related to the Bose gas, such as the ground state energy and condensate fraction. I will then present a construction of the solutions to the equation, and discuss some of their properties. |
| Date: | May 22, 2020 |
| Time: | 1:50pm |
| Location: | Zoom Seminar |
| Speaker: | Ilya Kachkovskiy, MSU |
| Title: | On spectral band edges of discrete periodic Schrodinger operators |
| Abstract: | We consider discrete Schrodinger operators on $\ell^2(\mathbb Z^d)$, periodic with respect to some lattice $\Gamma$ in $\mathbb Z^d$ of full rank. Our main goal is to study dimensions of level sets of spectral band functions at the energies corresponding to their extremal values (the edges of the bands). Suppose that $d\ge 3$ and the dual lattice $\Gamma’$ does not contain the vector $(1/2,…,1/2)$. Then the above mentioned level sets have dimension at most $d-2$. Suppose that $d=2$ and the dual lattice does not contain vectors of the form $(1/p,1/p)$ and $(1/p,-1/p)$ for all $p\ge 2$. Then the same statement holds (in other words, the corresponding level sets are finite modulo $\mathbb Z^d$). For all lattices that do not satisfy the above assumptions, there are known counterexamples of level sets of dimensions $d-1$. Part of the argument also implies a discrete Bethe-Sommerfeld property: if $d\ge 2$ and the dual lattice does not contain the vector $(1/2,…,1/2)$, then, for sufficiently small potentials (depending on the lattice), the spectrum of the periodic Schrodinger operator is an interval. Previously, this property was studied by Kruger, Embree-Fillman, Jitomirskaya-Han, and Fillman-Han. Our proof is different and implies some new cases. The talk is based on joint work with in progress with N. Filonov. |
