This is the regular Mathematical Physics Seminar at the Mathematics Department of the Pontificia Universidad Católica de Chile.
Speaker: Rafael Benguria, PUC
Title: Bounds on the maximum ionization of atoms
Place: Pontificia Universidad Católica, Facultad de Matemáticas, Campus San Joaquin, Sala 1Edificio Felipe Villanueva
Abstract:
In this talk I will present new bounds on the maximum ionization of a system of N boson particles interacting via the Coulomb potential in R3. This is joint work with Juan Manuel Gonzalez and Trinidad Tubino.
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.
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.
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.
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).
Speaker: Daniel J. Pons. Universidad Andrés Bello
Title: Métricas no-canónicas en Diff(S1)
Place: Pontificia Universidad Católica, Facultad de Matemáticas, Campus San Joaquin, Sala 5
Abstract:
Re-visitamos ideas de V. I. Arnold sobre grupos de difeomorfismos de variedades. Cuando la variedad subyacente es el círculo, estudiamos la geometría de tal grupo dotado con algunas métricas.
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.
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.
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.
Speaker: Horia Cornean. Aalborg University
Title: A Beals criterion for magnetic pseudo-differential operators proved with magnetic Gabor frames
Place: Pontificia Universidad Católica, Facultad de Matemáticas, Campus San Joaquin, Sala 5
Abstract:
First, we give a new proof for the Beals commutator criterion for non-magnetic Weyl pseudo-differential operators based on classical Gabor tight frames. Second, by introducing a modified ‘magnetic’ Gabor tight frame, we naturally derive the magnetic analogue of the Beals criterion originally considered by Iftimie-Mantoiu-Purice. This is joint work with Bernard Helffer (Nantes) and Radu Purice (Bucharest). https://doi.org/10.1080/03605302.2018.1499777