# 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.
2020-09-30
15:45 hrs.
Rafael Tiedra de Aldecoa. Pontificia Universidad Católica de Chile
Mourre theory for unitary operators in two Hilbert spaces and quantum walks on trees
Zoom Meeting ID: 93311125756, Password: FisMat
Abstract:

For unitary operators $U_0,U$ in Hilbert spaces ${\mathcal H}_0,{\mathcal H}$ and identification operator $J:{\mathcal H}_0\to{\mathcal H}$, we present results on the derivation of a Mourre estimate for $U$ starting from a Mourre estimate for $U_0$ and on the existence and completeness of the wave operators for the triple $(U,U_0,J)$. As an application, we determine spectral and scattering properties of a class of anisotropic quantum walks on homogenous trees of degree $3$ with evolution operator $U$. In particular, we establish a Mourre estimate for $U$, obtain a class of locally $U$-smooth operators, and prove that the spectrum of $U$ covers the whole unit circle and is purely absolutely continuous, outside possibly a finite set where $U$ may have eigenvalues of finite multiplicity. We also show that (at least) three different choices of free evolution operators $U_0$ are possible for the proof of the existence and completeness of the wave operators.

https://zoom.us/j/93311125756?pwd=Tk96UlNwNDJCaElIZVNCTWc2azQ0UT09
2020-05-20
15:45 hrs.
Asaf Levi Franco Arellano. Unam
Operadores de Sturm-Liouville Con Interacciones Puntuales
online seminar with zoom
Abstract:
En esta charla estudiaré la invarianza de los valores propios de operadores de Sturm-Liouville autoadjuntos con interacciones puntuales. En un ambiente probabilístico, dada una familia $H_\omega$ de esta clase de operadores mostraré que un punto es valor propio para toda $\omega$ o solo para un conjunto de $\omega$'s de medida cero.
2020-01-22
15:45 hrs.
Some isoperimetric problems in the Euclidean space with density
Abstract:
We will discuss the isoperimetric problem for factorized measures obtained as perturbations of the Gaussian and the anti-Gaussian, respectively. Among other things, we will show that some isoperimetric problems, for which balls centered at the origin are stable, have no solutions.Time permitting, some applications, like, for instance, Faber-Krahn type inequalities will be presented too. (Joint works with F. Brock and A. Mercaldo)
2019-12-04
15:45 hrs.
Diego Garcia. Pontificia Universidad Católica de Chile
Nematic Bulk Superconductivity In Presence Of An Electric Field
Sala 5
Abstract:
In this talk I will show the theoretical framework for the obtention of a nematic inhomogeneous superfluid model, in presence of an electric field, by using the Ginzburg Landau theory. In this context, I will conclude that the irrotational part of the superfluid current and the electric field influences the possibility of a Fréedericksz transition for the molecular alignment. Furthermore, under several assumptions, this transition is determined by a non trivial solution for the nematic phase of the system. Finally, I will suggest a set of mathematical open questions that could contribute to understand this problem. This is a joint work with Juan Pablo Borgna (Universidad de San Martin, Argentina) and Mariano De Leo (Universidad de General Sarmiento, Argentina).
2019-10-09
15:45 hrs.
Marcone Corrêa Pereira. Universidad de Sao Paulo
A nonlocal approach to spatial spread in thin structures
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 $\epsilon$-parameter family of open sets which squeezes to a lower dimension open set as $\epsilon \to 0$. The asymptotic behavior of the solutions is analyzed and the results are compared with classical situations to elliptic equations in thin domains.
2019-09-11
15:45 hrs.
Juan Felipe Lopez Restrepo. Universidad de los Andes, Colombia
Edge States and Selfadjoint Extensions in the Kitaev Chain
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.
2019-09-04
15:45 hrs.
Rafael Tiedra de Aldecoa. Pontificia Universidad Católica de Chile
Ruled Strips With Asymptotically Diverging Twisting
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). https://doi.org/10.1007/s00023-018-0684-4
2019-08-28
15:45 hrs.
Daniel J. Pons. Universidad Andrés Bello
Métricas no-canónicas en Diff(S1)
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.
2019-08-21
15:45 hrs.
Marouane Assal. Pontificia Universidad Católica de Chile
A double well problem for a system of Schrödinger operators with energy-level crossing
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.

This is a joint work with Setsuro Fujiie (Ritsumeikan University, Kyoto, Japan).
2019-08-14
15:45 hrs.
Local energy decay for the periodic damped wave equation
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.
2019-07-24
15:45 hrs.
Fabian Belmonte. Universidad Católica del Norte
Canonical Quantization of Constants of Motion
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.
2019-06-05
15:45 hrs.
Horia Cornean. Aalborg University
A Beals criterion for magnetic pseudo-differential operators proved with magnetic Gabor frames
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
2019-05-29
15:45 hrs.
Andrés Fernando Reyes Lega. Universidad de los Andes (Colombia)
Emergent gauge symmetries, quantum operations and anomalies
Sala 5
Abstract:
The Gelfand-Naimark-Segal (GNS) construction is a fundamental tool for the study of the representation theory of operator algebras. It also plays a prominent role in the algebraic approach to quantum field theory. In this talk I will discuss some examples of applications of the algebraic approach to quantum physics to systems with a finite number of degrees of freedom. I will illustrate how the GNS construction naturally leads to interesting connections between gauge symmetries, anomalies and quantum-information concepts like entanglement entropy and quantum operations.
2019-05-15
15:45 hrs.
Massimo Moscolari. Sapienza University of Rome
Beyond Diophantine Wannier diagrams: gap labelling for Bloch-Landau Hamiltonians
Sala 5
Abstract:
In 1978 Wannier discovered a Diophantine relation expressing the integrated density of states of a gapped group of bands of the Hofstadter Hamiltonian as a linear function of the magnetic field flux with integer slope. I will show how to extend this relation to a gap labelling theorem for any 2D Bloch-Landau Hamiltonian operator and to certain non-covariant systems having slowly varying magnetic fields. The integer slope will be interpreted as the Chern character of the projection onto the space of occupied states. The talk is based on a joint work with H. Cornean and D. Monaco.
2019-04-24
15:45 hrs.
Svetlana Jitomirskaya. University of California, Irvine
Cantor spectrum of a model of graphene in magnetic field
Sala 5
Abstract:
We consider a quantum graph as a model of graphene in magnetic fi elds and give a complete analysis of the spectrum, for all constant fluxes. In particular, we show that if the reduced magnetic flux through a honeycomb is irrational, the continuous spectrum is an unbounded Cantor set of Lebesgue measure zero and Hausdorff dimension bounded by 1/2.

Based on joint works with S. Becker, R. Han, and also I. Krasovsky.
2019-04-17
15:45 hrs.
Walter de Siqueira Pedra. University of São Paulo
Thermodynamical Stability and Dynamics of Lattice Fermions with Mean-Field Interactions
Sala 5
Abstract:
For lattice fermions we study the thermodynamic limit of the time evolution of observables when the corresponding finite-volume Hamiltonians contain mean-field terms (like, e.g., the BCS model). It is well-known that, in general, this limit does not exist in the sense of the norm of observables, but may exist in the strong operator topology associated to a well-chosen representation of the algebra of observables. We proved that this is always the case for any cyclic representation associated to an invariant minimizer of the free energy density, if the Hamiltonians are invariant under translations. Our proof uses previous results on the structure of states minimizing the free energy density of mean-field models along with Lieb-Robinson bounds for the corresponding families of finite-volume time evolutions. This is a joint work with Jean-Bernard Bru, Sébastien Breteaux and Rafael Miada.
2019-04-03
15:45 hrs.
Jorge Antezana. National University of la Plata
Quasicrystals and Fourier analysis
Sala 5
Abstract:
Quasicrystals are non-periodic structures discovered by Shechtman in 1984 (see [Sh]). Nowadays, one of the best mathematical descriptions quasicrystals are the so called "model sets". These sets were introduced by Meyer in [M], many years before the discovery of Shechtman. In that moment, one of the aims of Meyer was to study approximation of algebraic characters by continuous ones in locally compact abelian groups (see also [L]).

Recently, important applications of quasicrystals to Fourier Analysis have been found (see [MM], [GL], [LO], [AACM] ). In this talk we will discuss some of these applications, making focus in those related with problems of sampling and interpolation in Paley Wiener spaces.

[AACM]  E. Agora, J. Antezana, C. Cabrelli, Existence of quasicrystals and universal stable sampling and interpolation in LCA groups, to appear in Trans. Amer. Math. Soc.

[GL] S. Grepstad, N. Lev,  Multi-tiling and Riesz bases. Adv. Math. 252 (2014), 1-6.

[L] J. C. Lagarias, Mathematical quasicrystals and the problem of diffraction. Directions in mathematical quasicrystals,  CRM Monogr. Ser., 13, Amer. Math. Soc., Providence (2000) 61-93.

[LO] N. Lev, A. Olevskii, Quasicrystals and Poisson's summation formula, Invent. math. 200 (2015), 585-606.

[MM] B. Matei, Y. Meyer, Simple quasicrystals are sets of stable sampling, Complex Var. Elliptic Equ. 55 (2010), 947-964.

[M] Y. Meyer, Algebraic Numbers and Harmonic Analysis, (1970) North Holland.

[Sh] D. Shechtman, I. Blech, D. Gratias, J.W. Cahn,  Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 53 (1984) 1951-1953.
2019-03-13
15:45 hrs.
Monika Anna Winklmeier. Universidad de los Andes
Estimates for eigenvalues in gaps of the essential spectrum
Sala 5
Abstract:
In this talk I will show how bounds for eigenvalues in gaps of the essential spectrum of a linear operator can be obtained. The main example will be a one-dimensional Dirac type operator.
2019-03-06
15:45 hrs.
Christian Jaekel. University of São Paulo
On reflection positivity, modular localisation and Connes cocycles
Sala 5
Abstract:
The unitary irreducible representations of the Lorentz group carry an intrinsic notion of localisation on de Sitter space, known as modular localisation. An extension of Araki’s perturbation theory of modular automorphisms can be used to define interacting representations of the Lorentz group, as well as the corresponding Haag-Kastler nets. The analyticity properties of the correlation functions allow us to extend these theories to “nets" of (non-abelian) von Neumann algebras on the sphere. Reflection positivity can be used to recover the interacting quantum (field) theories on the de Sitter space from the sphere. Explicit examples are scalar bosons with polynomial or exponential interactions in 1+1 space-time dimensions, but our aim is to classify all interacting quantum theories compatible with the space-time symmetries. The Minkowski space limit is the limit of space-time curvature to zero, which is well-behaved on the level of local von Neumann algebras.
2018-12-19
15:45 hrs.
Jean Bellissard. Georgia Institute of Technology
Viscosity of liquids: A simplistic but effective model
Sala 5
Abstract:
Using a new degree of freedom called "anankeon" we design a Markov process describing the competition with phonons. The viscosity can be computed analytically. It can be shown that the behavior in temperature follows experimental results.