Schedule and abstracts

Abstract: This talk will focus on Dirac operators that emerge when studying macroscopic transport between topological insulators. I will analytically construct canonical edge states: coherent states that propagate along interfaces, but do not admit natural counter-propagating companions. I will illustrate the results with various numerical simulations.

Abstract: The spectral gap conjecture is a well known and widely believed conjecture in mathematical physics concerning the structure of the Bloch variety (dispersion relation) of periodic operators. The Bloch variety of a discrete operator is algebraic, inviting methods from algebraic geometry to their study. Motivated by this conjecture, this talk will introduce a bound on the number of critical points of the dispersion relation for discrete periodic operators, and provide a general criterion for when this bound is achieved. We also present a class of periodic graphs for when this criteria is satisfied for Laplace-Beltrami operators. This is joint work with Frank Sottile.

Abstract: We study the Heisenberg XXZ chain with local spin J ($J \in \{1/2,1,3/2,...\}$) with background magnetic field. As in the spin-1/2 case, it s possible to rewrite the Hamiltonian as a direct sum of $N$-particle Schr\"odinger operators with attractive interaction. We will present a bound on the entanglement entropy for states of arbitrarily high (but fixed) energy. After this, we discuss localization results under the presence of a random magnetic background field, which follow from a modification of work by Elgart and Klein in the spin-1/2 case. This is joint work with Fisher, Klein (Localization) and Ogunkoya (Entanglement Entropy).

Abstract: Using the variational characterization of the principal (i.e., smallest) eigenvalue below the essential spectrum of a lower semibounded self-adjoint operator, we prove strict domain monotonicity (with respect to changing the finite interval length) of the principal eigenvalue of the Friedrichs extension T_F of the minimal operator for regular four-coefficient Sturm-Liouville differential expressions.

As a consequence of the strict domain monotonicity of the principal eigenvalue of T_F in the regular case, and on the basis of oscillation theory in the singular context, we characterize all lower bounds of T_F as those real energies for which the corresponding differential equation has a strictly positive solution on the underlying interval (a,b).

Abstract: I will discuss my proof of localization for the Anderson model with discrete disorder along with ideas for an extension to the case of Bernoulli disorder (joint work with S. Mayboroda).

Abstract: I will discuss some recent results on spectral theory of Schrödinger operators over circle maps including circle diffeomorphisms and circle diffeomorphisms with singular points where the derivative vanishes (critical circle maps) or has a jump discontinuity (maps with breaks).

Abstract: We explain the sharp Lorentz resonances in plasmonic crystals that consist of 2D nano dielectric inclusions as the interaction between resonant material properties and geometric resonances of electrostatic nature. One example of such plasmonic crystals are graphene nanosheets that are periodically arranged within a non-magnetic bulk dielectric. We derive an analytic formula for the Lorentz resonances which decouples the geometric contribution and the frequency dependence. This formula comes rigorously from the corrector equation in the process of homogenization, and it can be used for efficient computation. This is joint work with Robert Lipton and Matthias Maier.

Abstract: In this talk, I will present joint work with Benjamin Eichinger and Brian Simanek: a new approach to universality limits for orthogonal polynomials on the real line which is completely local and uses only the boundary behavior of the Weyl m-function at the point. We show that bulk universality of the Christoffel--Darboux kernel holds for any point where the imaginary part of the m-function has a positive finite nontangential limit. This approach is based on studying a matrix version of the Christoffel--Darboux kernel and the realization that bulk universality for this kernel at a point is equivalent to the fact that the corresponding m-function has normal limits at the same point. Our approach automatically applies to other self-adjoint systems with 2x2 transfer matrices such as continuum Schrodinger and Dirac operators. We also obtain analogous results for orthogonal polynomials on the unit circle.

Abstract: In 1961, Birman proved a sequence of inequalities valid for functions in C_{0}^{n}((0,\infty)) containing the classical (integral) Hardy inequality and the well-known Rellich inequality, and much effort has been made in improving and extending these inequalities with weights and singular logarithmic refinement terms.

In this talk, we discuss the optimality of the Birman inequalities. In particular, we introduce a new proof of the power-weighted Birman inequalities without refinement terms, using a modified variable transformation in integrals first studied by Hartman and Müller-Pfeiffer.

This is based on recently published work with Fritz Gesztesy, Lance Littlejohn, and Michael Pang.

Abstract: We consider two families of quantum spin chains with O(n)-invariant nearest-neighbor interactions and discuss the ground state phase diagram of this class of models. Using a graphical representation for the partition function, we provide a proof of a spectral gap above the ground states and spontaneous breaking of the translation symmetry for an open region in the phase diagram, for all sufficiently large values of n. (Joint work with Jakob Bjoernberg, Peter Muehlbacher, and Daniel Ueltschi).

Abstract: Reducibility of the Fermi surface for a periodic operator is a key for the existence of embedded eigenvalues caused by a local defect. We consider a bilayer quantum graph model for a quantum system subject to a magnetic field, stacked graphene being a particular example. Some techniques from non-magnetic operators extend to magnetic ones, but the magnetic case is more complex because a typical magnetic operator on a periodic graph is not periodic.

Abstract: In an inverse boundary problem, one seeks to determine the coefficients of a PDE inside a domain, describing internal properties, from the knowledge of boundary values of solutions of the PDE, encoding boundary measurements. Applications of such problems range from medical imaging to non-destructive testing. In this talk, starting with the fundamental Calderon inverse conductivity problem, we shall first discuss inverse boundary problems for first order perturbations of biharmonic operators in the setting of compact Riemannian manifolds with boundary. Specifically, we shall present a global uniqueness result as well as a reconstruction procedure for the latter inverse boundary problem on conformally transversally anisotropic Riemannian manifolds of dimensions three and higher. Finally, we shall also discuss briefly inverse boundary problems for nonlinear magnetic Schroedinger operators on a compact complex manifold, illustrating the recent insight that the presence of nonlinearity may help when solving inverse problems.

Abstract: Quantum walks are quantum analogies of the classic random walks. In this talk, we will discuss some recent results, in particular Anderson localization for all Diophantine frequencies, for quantum walk models in magnetic and electric fields.

Abstract: In this talk, we will discuss recent work examining the transport properties of the class of limit-periodic continuum Schrödinger operators whose potentials are approximated exponentially quickly by a sequence of periodic functions. For such an operator $H$, and $X_H(t)$ the Heisenberg evolution of the position operator, we show the limit of $\frac{1}{t}X_H(t)\psi$ as $t\to\infty$ exists and is nonzero for $\psi\ne 0$ belonging to a dense subspace of initial states which are sufficiently regular and of suitably rapid decay. This is viewed as a particularly strong form of ballistic transport, and this is the first time it has been proven in a continuum almost periodic non-periodic setting. In particular, this statement implies that for the initial states considered, the second moment grows quadratically in time.

Abstract: The twisted bilayer graphene is famous for the phenomenon that when two sheets of graphene are twisted against each other at certain magic angles, the electronic structure undergoes a transition from the Mott-insulating states to the unconventional superconducting states. In this talk, I will talk about the magnetic response of such materials using an effective magnetic Bistritzer-MacDonald model. We derive the asymptotic expansion of the DOS in a strong magnetic field. The explicit expansion allows us to study the magnetic oscillations and quantum Hall effects. In particular, we will see how the two different tunnelings, AB'/BA'-tunnelings and AA'/BB'-tunnelings, contribute to the DOS and magnetic responses in very different ways. This is joint work with Simon Becker and Jihoi Kim.