Seminario FisMat

El objetivo de este seminario es de reunir, de la manera la mas amplia posible, investigadores y estudiantes de la comunidad chilena e internacional alrededor de las diversas temáticas de física matemática. Profesores, investigadores jóvenes, así como estudiantes, son los bienvenidos como expositores.

Los miércoles, a las 15:45 hrs, sala 5 de la Facultad de Matemáticas.

Organización: Olivier Bourget, Giuseppe De Nittis, Christian Sadel, Edgardo Stockmeyer, Rafael Tiedra de Aldecoa.
2018-04-04
15:45 hrs.
Rafael Tiedra de Aldecoa. Pontificia Universidad Católica de Chile
Quantum time delay for unitary propagators
Sala 5
Abstract:
We give the definition of quantum time delay in terms of sojourn times for unitary propagators in a two-Hilbert spaces setting. We prove that this time delay defined in terms of sojourn times (time-dependent definition) exists and coincides with the expectation value of a unitary analogue of the Eisenbud-Wigner time delay operator (time-independent definition). Our proofs rely on a new summation formula relating localisation operators to time operators and on various tools from functional analysis such as Mackey's imprimititvity theorem, Trotter-Kato Formula and commutator methods for unitary operators. Joint work with Diomba Sambou (PUC).
https://doi.org/10.1142/S0129055X19500181
2018-03-28
15:45 hrs.
Diomba Sambou. Pontificia Universidad Católica de Chile
Complex eigenvalues for a non-self-adjoint Dirac operator
Sala 5
Abstract:
We will consider a 2d Dirac operator with constant magnetic field perturbed by non-self-adjoint potentials. It is well known that when it is perturbed by certain self-adjoint potentials, then, there is creation and accumulation of real eigenvalues near every point of its essential spectrum given by a set of degenerate isolated eigenvalues called the Landau levels. Recently, similar results have been proved for Schrödinger operators perturbed by non-self-adjoint perturbations showing the existence of complex-valued potentials generating infinitely many non-real eigenvalues accumulating at every point of [0,+∞). We will present a similar result for the 2d Dirac operator above, showing the existence of non-self-adjoint perturbations generating infinitely many non-real eigenvalues accumulating at every Landau level.
2018-03-21
15:45 hrs.
Nicolas Raymond. University of Rennes 1
Survey on the semiclassical magnetic Laplacian
Sala 5
Abstract:
In the first part of the talk, I will describe recent advances about the description of the discrete spectrum of the magnetic Laplacian, in the semiclassical limit. In the second part, I will describe two results in two dimensions, essentially in the case of a non-degenerate magnetic well: the Birkhoff normal forms and their applications (collaboration with S. Vu Ngoc) and the magnetic WKB constructions (collaboration with Y. Bonthonneau).
2018-03-14
15:45 hrs.
Timo Weidl. University of Stuttgart
The edge resonance in elastic media with zero Poisson coefficient
Sala 5
Abstract:
A two-dimensional elastic semistrip with stress-free boundary conditions and zero Poisson coefficients has an embedded eigenvalue on top of the continuous spectrum. This effect is known as edge resonance. For an infinite plate of finite thickness with a drilling hole $(R^2\setminus\Omega)\times I$ actually infinitely many edge resonances will occur. This is related to the spectral problem of perturbations of symbols with strongly degenerated minima, which also appear in BCS theory. I give an overview on some of our results in this area, which still poses a number of mathematical challenges.
2017-12-20
15:45 hrs.
Michael Loss. Georgia Institute of Technology
Entropy decay for the Kac master equation
Sala 5
Abstract:
The Kac master equation models the behavior of a large number of randomly colliding particles. Due to its simplicity it allows, without too much pain, to investigate a number of issues. E.g., Mark Kac, who invented this model in 1956, used it to give a simple derivation of the spatially inhomogeneous Boltzmann equation. One important issue is the rate of approach to equilibrium, which can be analyzed in various ways, using, e.g., the gap or the entropy. Explicit entropy estimates will be discussed for a Kac type master equation modeling the interaction of a finite system with a large but finite reservoir. This is joint work with Federico Bonetto, Alissa Geisinger and Tobias Ried.
2017-12-06
15:45 hrs.
Daniel Remenik. Cmm, Universidad de Chile
The KPZ fixed point
Sala 5
Abstract:
I will describe the construction and main properties of the KPZ fixed point, which is the scaling invariant Markov process conjectured to arise as the universal scaling limit of all models in the KPZ universality class, and which contains all the fluctuation behavior seen in the class. The construction follows from an exact solution of the totally asymmetric exclusion process (TASEP) for arbitrary initial condition. This is joint work with K. Matetski and J. Quastel.
2017-11-29
15:45 hrs.
Giovanni Landi. Trieste University
Noncommutative products of Euclidean spaces
Sala 5
Abstract:
We present natural families of coordinate algebras of noncommutative products of Euclidean spaces. These coordinate algebras are quadratic ones associated with an R-matrix which is involutive and satisfies the Yang-Baxter equations. As a consequence they enjoy a list of nice properties, being regular of finite global dimension. Notably, we have eight-dimensional noncommutative euclidean spaces which are particularly well behaved and are parametrised by a two-dimensional sphere. Quotients include noncommutative seven spheres as well as noncommutative "quaternionic tori". There is invariance for an action of SU(2) x SU(2) in parallel with the action of U(1) x U(1) on a "complex" noncommutative torus which allows one to construct quaternionic toric noncommutative manifolds. Additional classes of solutions are disjoint from the classical case.
2017-11-22
15:45 hrs.
Hanne Van Den Bosch. Universidad de Chile
Non-existence of minimizers for the TFDW model
Sala 5
Abstract:
It is a well-known fact in physics that a neutral atom can bind at most one or two additional electrons. But proving such a bound on the maximum positive ionization is still an open problem in many-body quantum mechanics. Recently, we were able to prove such a bound in several approximate models for the full quantum theory: the Thomas-Fermi-Dirac-von Weiszäcker density functional and Müller’s density matrix functional. In this talk, I will give an overview of the known results and techniques, illustrate our method in a simplified model and give the basic ideas behind the proof. This talk is based on joint work with R. L. Frank and P. T. Nam (LMU Munich).
2017-11-15
15:45 hrs.
Siegfried Beckus. Technion (Israel Institute of Technology)
Spectral Approximation of Schrödinger Operators
Sala 5
Abstract:
The first quasicrystals where discovered by D. Shechtman in the year 1984. From the mathematical point of view, the study of the associated Schrödinger operators turns out to be a challenging question. Up to know, we can mainly analyze one-dimensional systems by using the method of transfer matrices. In 1987, A. Tsai et al. discovered a quasicrystalline structure in an Aluminum-Copper-Iron composition. By changing the concentration of the chemical elements, they produce a stable quasicrystaline structure by an approximation process of periodic crystals. In light of that it is natural to ask whether Schrödinger operators related to aperiodic structures can be approximated by periodic ones while preserving spectral properties. The aim of the talk is to provide a mathematical foundation for such approximations. In the talk, we develop a theory for the continuous variation of the associated spectra in the Hausdorff metric meaning the continuous behavior of the spectral gaps. We show that the convergence of the spectra is characterized by the convergence of the underlying dynamics. Hence, periodic approximations of Schrödinger operators can be constructed by periodic approximations of the dynamical systems which we will describe along the lines of an example.
2017-11-08
15:45 hrs.
Jean Bellissard. Georgia Institute of Technology / Westfälische Wilhelms-Universität
Electronic Transport in Aperiodic Solids
Sala 5
Abstract:
The concept of transport and transport coefficient will be introduced with an emphasis upon dissipation mechanisms. The Kubo formula will be derived both for perfect metals (Drude formula) and for aperiodic media. Emil Prodan designed a code based upon the non commutative approach to describing electrons in aperiodic solids that makes the Kubo formula computable in a numerically stable way. A series of numerical results related to the Quantum Hall Effect will be presented. They show that several accurate predictions can be made, unseen before.
Coherent transport, transport of charges occurring without dissipation, and can be observed experimentally at very low temperature, will be introduced as well as the scaling exponents which characterize it. The competition between coherent transport and dissipation has consequences on the low temperature behavior of the electric conductivity. It leads to an anomalous Drude formula representing well the strange behavior of aperiodic solids, like quasicrystals or doped semiconductors, at low temperature. An emphasis upon the case of quasicrystals will be provided, with two different regimes, based on an argument by Thouless, which may be an explanation for various observations made so far by experimentalists.