Seminar Spectral Theory and PDE

Seminar Spectral Theory and PDE, October 25, 13:00

Speaker: Nora Doll. Friedrich-Alexander University of Erlangen

Title: Connecting Real index pairings to spectral flows

Place: PUC Chile, Campus San Joaquin, Fac. Matematicas, Sala 2

Abstract:
Object of this talk are index pairings of a projection and an unitary where both, the projection and the unitary fulfill Real symmetry relations. For a given combination of  symmetries the Noether index of the pairing vanishes, but there may be a Z_2-index given by the dimension of the kernel, modulo 2. Aim of this talk is to give an interpretation of these Z_2-indices as spectral flows. Two applications are given, one concerning topological insulators, the other concerning the Spectral Localizer.

seminar FisMat

FisMat Seminar, October 23, 15:45

Speaker: Max Lein. Tohoku University

Title: On the Topological Nature of Electromagnetic Surface Modes at Metal-Dielectric Interfaces

Place: Pontificia Universidad Católica, Facultad de Matemáticas, Campus San Joaquin, Sala 5

Abstract:
Phenomena that can be linked to the “topology of the system” have become quite ubiquitous and popular in physics, as they tend to be very robust to perturbations. Bulk-boundary correspondences make this mathematically precise. In a recent publication we have identified the presence of electromagnetic surface waves at metal-dielectric interfaces of a given polarization as a topological phenomenon. More precisely, we have shown that we can predict the existence of these surface modes by a hitherto unknown type of bulk-boundary correspondences that has no quantum analog. From a mathematical point of view, these bulk-boundary correspondences should still be regarded as a conjecture, since our work rests on explicit solutions to Maxwell’s equations for planar interfaces.

In this talk, I will explain how we have arrived at these conjectures despite the absence of a complete mathematical theory explaining the underlying mechanism. And I will propose a mathematical framework in which this bulk-boundary correspondence might be made rigorous. Three aspects are particularly intriguing: from the conceptual point of view, this is to our knowledge the first topological effect that is explained not by the analog of the hamiltonian, but by the topology of another conserved quantity — helicity. Secondly, a mathematical explanation involves Krein-selfadjoint operators on Krein spaces (as opposed to selfadjoint operators on Hilbert spaces). Roughly speaking, Krein spaces are Hilbert spaces equipped with a second, indeterminate inner product. And lastly, even the new bulk classification of non-hermitian topological insulators obtained independently by Zhou and Lee as well as Kawabata et al. does not correctly predict the topology of the system.

seminar FisMat

FisMat Seminar, October 9, 15:45

Speaker: Marcone Corrêa Pereira. Universidad de Sao Paulo

Title: A nonlocal approach to spatial spread in thin structures

Place: Pontificia Universidad Católica, Facultad de Matemáticas, Campus San Joaquin, Sala 5

Abstract:
In this talk we discuss an approach to considerer spatial spread in N-dimensional thin structures.  We introduce equations with nonlocal dispersal and defined in tight domains contrasting it with its corresponding local diffusion equation with Neumann and Dirichlet boundary conditions. Here the thin structure effect is modeled by an ϵ-parameter family of open sets which squeezes to a lower dimension open set as ϵ→0. The asymptotic behavior of the solutions is analyzed and the results are compared with classical situations to elliptic equations in thin domains.

seminar FisMat

FisMat Seminar, September 11, 15:45

Speaker: Juan Felipe Lopez Restrepo. Universidad de los Andes, Colombia

Title: Edge States and Selfadjoint Extensions in the Kitaev Chain

Place: Pontificia Universidad Católica, Facultad de Matemáticas, Campus San Joaquin, Sala 5

Abstract:
In this seminar, finite discrete Kitaev chain shall be presented with its topological phase transition. A continuum limit model is derived and the diagonalization problem of the resulting bilinear hamiltonian is translated in choice of selfad-joint extensions for a one dimensional Dirac operator, which is given by a set of possible boundary conditions. It is shown that the previous result coincides with the application of the Araki’s self dual formalism and its connection with edge sates is discussed. Joint work with A. F. Reyes.

seminar FisMat

FisMat Seminar, September 4, 15:45

Speaker: Rafael Tiedra de Aldecoa. Pontificia Universidad Católica de Chile

Title: Ruled Strips With Asymptotically Diverging Twisting

Place: Pontificia Universidad Católica, Facultad de Matemáticas, Campus San Joaquin, Sala 5

Abstract:
We consider the Dirichlet Laplacian in a 2-dimensional strip composed of segments translated along a straight line with respect to a rotation angle with velocity diverging at infinity. We show that this model exhibits a “raise of dimension” at infinity leading to an essential spectrum determined by an asymptotic 3-dimensional tube of annular cross section. If the cross section of the asymptotic tube is a disc, we also prove the existence of discrete eigenvalues below the essential spectrum. Joint work with David Krejcirik (Prague).

seminar FisMat

FisMat Seminar, August 21, 15:45

Speaker: Marouane Assal. Pontificia Universidad Católica de Chile

Title: A double well problem for a system of Schrödinger operators with energy-level crossing

Place: Pontificia Universidad Católica, Facultad de Matemáticas, Campus San Joaquin, Sala 5

Abstract:
We study the existence and the asymptotic distribution of the eigenvalues of a 2*2 semiclassical system of coupled Schrödinger operators, in the case where the two electronic levels (potentials) cross at some real point and each of them admits a simple well. Considering energy levels above that of the crossing, we give the asymptotics of the eigenvalues close to such energies. In the case of symmetric wells, eigenvalues splitting occurs and we give a precise estimate of it.

seminar FisMat

FisMat Seminar, August 14, 15:45

Speaker: Julien Royer. Universidad de Toulouse

Title: Local energy decay for the periodic damped wave equation

Place: Pontificia Universidad Católica, Facultad de Matemáticas, Campus San Joaquin, Sala 5

Abstract:
In this talk, we will discuss the local (or global) energy decay for the wave equation with damping at infinity. We are in particular interested in the case of a periodic (or asymptotically periodic) setting. We will mainly describe the contribution of low frequencies and observe that it behaves like the solution of some heat equation. We will see how this emerges from the spectral analysis of the damped wave equation.

seminar FisMat

FisMat Seminar, July 24, 15:45

Speaker: Fabian Belmonte. Universidad Católica del Norte

Title: Canonical Quantization of Constants of Motion

Place: Pontificia Universidad Católica, Facultad de Matemáticas, Campus San Joaquin, Sala 5

Abstract:
It is well known that Weyl quantization does not intertwine the Poisson bracket of two functions with the commutator of the corresponding operators (Groenewold- van Hove’s no go theorem). The latter suggest that Weyl quantization does not preserve the constants of motion of every given Hamiltonian, however, there are very important examples where it does so. In this talk we are going to approach the following problems:
a) Is it possible to determine the Hamiltonians for which a given canonical quantization preserves its constants of motion? We will give an interesting criteria partially answering this question in terms of the Wigner transform. We will give some important examples as well.
b) Conversely, is it possible to construct a canonical quantization preserving the constants of motion of a prescribed Hamiltonian? Under certain conditions, we will show a construction of such quantization based in the structural analogy between the description of classical and quantum constants of motion.

Seminar Spectral Theory and PDE

Seminar Spectral Theory and PDE, June 27, 17:00

Speaker: Leonid Parnovski. University College London

Title: Floating mats and sloping beaches: spectral asymptotics of the Steklov problem on polygons.

Place: PUC Chile, Campus San Joaquin, Fac. Matematicas, Sala 2

Abstract:
I will discuss recent results (joint with M.Levitin, I.Polterovich and D.Sher) on the asymptotic behaviour of Steklov eigenvalues on polygons and other two-dimensional domains with corners. The answer is completely unexpected and depends on the arithmetic properties of the angles.