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 14:30 hrs, 
Organización:  Giuseppe De NittisGregorio Moreno, Amal Taarabt

 


2022-12-07
14:30hrs.
Tobias Ried. Max Planck Institute
Cwikel's bound reloaded
Sala 1 - Facultad de Matemáticas
Abstract:
The Cwikel-Lieb-Rozenblum (CLR) inequality is a semi-classical bound on the number of bound states for Schrödinger operators. Of the rather distinct proofs by Cwikel, Lieb, and Rozenblum, the one by Lieb gives the best constant, the one by Rozenblum does not seem to yield any reasonable estimate for the constants, and Cwikel’s proof is said to give a constant which is at least about 2 orders of magnitude off the truth.
In this talk I will give a brief overview of the CLR inequality and present a substantial refinement of Cwikel’s original approach which leads to an astonishingly good bound for the constant in the CLR inequality. Our proof is quite flexible and leads to rather precise bounds for a large class of Schrödinger-type operators with generalized kinetic energies. Moreover, it highlights a natural but overlooked connection of the CLR bound with bounds for maximal Fourier multipliers from harmonic analysis. (joint work with D. Hundertmark, P. Kunstmann, and S. Vugalter)
2022-11-23
14:30hrs.
Danilo Polo. UC
Topological quantization of interface currents in magnetic quarter-plane systems
Sala 1 - Facultad de Matemáticas
Abstract:
A magnetic interface is a thin region of the space that separates the material in two parts subjected to perpendicular uniform magnetic fields of distinct intensity. It is well-known that interfaces may lead to surface modes which can carry edge currents. The aim of this talk is to show (in the tight-binding approximation) the topological quantization of such currents for magnetic quarter-plane systems, and by using K-theory, we derive bulk-interface dualities. If time allows, I will state the necessary conditions to produce topological non-trivial corner states in these systems.
Joint work with: Giuseppe De Nittis
2022-11-16
14:30hrs.
Avelio Sepúlveda. U de Chile
Relaciones entre el campo libre Gaussiano y el modelo de spins O(N)
Sala 1 - Facultad de Matemáticas
Abstract:
En esta presentación, se discutirá la correlación de dos puntos del modelo de spin O(3).  Siguiendo las ideas de Patrascioiu y Seiler, expresamos la correlación como una constante por la probabilidad que ambos puntos pertenezcan a una componente conexa de un modelo de percolación. Hacemos esto para mostrar que uno de los argumentos de estos autores puede ser contradecido construyendo un modelo $XY$ en un ambiente aleatorio que satisface lo siguiente: El modelo tiene baja temperatura excepto en un pequeño conjunto que no percola; y la función a dos puntos decrece exponencialmente. Es instrumental para estos resultados comprender la relación entre el campo libre Gaussiano y los modelos de spin O(N).
Trabajo en conjunto con Juhan Aru y Christophe Garban. 
2022-10-26
14:30hrs.
Giuseppe de Nittis. UC
The Magnetic Spectral Triple: Applications and Open Questions
Sala 1 - Facultad de Matemáticas
Abstract:
Since the early works by Bellissard, non-commutative geometry (NCG) has proved to be an excellent tool for the analysis of the quantum Hall effect (QHE), and more in general for the study of the topological phases of matter. The central object of the Bellissard's NCG for the QHE is a spectral triple designed to deal with tight-binding operators. In this talk we will present a new spectral triple suitable to treat  continuous magnetic operators. We will show how the QHE in the continuous can be described inside this new NCG. Certain possible new applications, along with some related open questions, will be also presented. Joint work with: F. Belmonte & M. Sandoval
2022-09-07
15:30hrs.
Bruno de Mendonça Braga . Puc-Rio
Embeddings of von Neumann algebras into uniform Roe algebras
Sala 1 - Facultad de Matemáticas
Abstract:
Given a uniformly locally finite metric space $X$, its uniform Roe algebra, denoted by $C^*_u(X)$, is a $C^*$-algebra of bounded operators on the Hilbert space $\ell_2(X)$ which captures the large scale geometry of $X$. This algebra was introduced by John Roe in 1988 and it has since become a topic of interest to researchers in many different fields such as operator algebras, geometric group theory, and mathematical physics. As for the latter, uniform Roe algebras have recently started to be used as a framework in mathematical physics to study the classification of topological phases. In this talk, we will discuss some recent developments about the structure of $C^*_u(X)$. More precisely, we discuss which von Neumann algebras can be found inside $C^*_u(X)$. 
2022-08-31
15.30hrs.
Pablo Miranda. Usach
Resonancias cerca de umbrales para operadores de Schrödinger discretos
Sala 1 - Facultad de Matemáticas
Abstract:
En esta charla consideramos versiones generalizadas de operadores Schrödinger definidos en $\mathbb{Z}\otimes\mathbb{C}^N$. Cerca de los umbrales del espectro estudiamos la distribución de las resonancias de nuestros operadores. La cantidad de resonancias cerca de cada umbral es finito y nuestro primer resultado es obtener el número exacto de estas. Además, obtenemos una descripción precisa de la localización de las resonancias, en términos de cúmulos cerca de ciertos puntos en el plano complejo. Estos resultados son bastante naturales pero no aparecen en la literatura físico-matemática.
En la segunda parte, consideramos operadores definidos en $\mathbb{Z}\otimes\mathbb{Z}$ pero con restricciones que inducen a las partículas a moverse de manera “horizontal”. Estos operadores tienen una estructura similar a los de la primera parte, pero presentan un fenómeno de acumulación de resonancias que nosotros describimos a través del comportamiento asintótico de la función de conteo de estas.
Parte de nuestro resultados son válidos para operadores no auto-adjuntos. Este es parte de un trabajo conjunto con Marouane Assal, Olivier Bourget y Diomba Sambou.
2022-07-28
15:30hrs.
Mircea Petrache|. UC
Building examples of graphs that allow infinitely many sharp isoperimetric shapes
Sala 1
Abstract:
Discrete isoperimetric shapes are configurations that reach equality in the edge-isoperimetric inequality among subsets of a fixed graph. Equivalently, the question is to find the shape of the best crystal grain, given the crystal structure of a material. Equality in the discrete edge-isoperimetric inequality is hard to achieve, and the ambient graphs which have infinite families of discrete isoperimetric shapes are rare: Our goal is to build a large class of examples. We start from the "macroscopic" or "continuum" isoperimetric problem, with two approaches, one via PDE and one via Optimal Transport. We build a new discrete strategy which combines the two approaches. Our strategy poses several nice new challenges, and it highlights the close link between semidiscrete optimal transport and convexity. In this introductory talk, I describe what new classes of examples we find, and also some mysterious directions still to be explored.
2022-06-22
11:30hrs.
Walter Alberto de Siqueira Pedra . Universidad de São Paulo
Equilibrium States of Many-Body (Fermion) Systems with Long-Range Interactions
Sala 1
Abstract:
We present our results on infinite volume equilibrium states of many-fermion systems on the lattice with mean-field interactions ("Non-cooperative Equilibria of Fermi Systems with Long-Range Interactions", Memoirs of the AMS, 2013) in relation to our recent contributions on Kac limits for equilibrium states ("From Short-Range to Mean-Field Models in Quantum Lattices", arXiv:2203.01021): We recently proved that under very general conditions such long-range limits of equilibria of short-range models are equilibria of some naturally associated mean-field model. This is reminiscent of well-known works of Penrose and Lebowitz (1966, classical case), and Lieb (1971, quantum), on the Kac limit of the pressure in the thermodynamic limit.
2022-06-09
15:30hrs.
Edgardo Stockmeyer. Pontificia Universidad Católica - Facultad de Física
Sobre la estabilidad de la ecuación de Dirac no-lineal en el modelo de Soler
Edificio Felipe Villanueva
Abstract:
Consideramos soluciones de tipo "onda estacionaria" en la ecuación de Dirac de dimensionalidad 1+1
en el modelo de Soler. La ecuación resultante tiene una no linealidad de tipo masa. En contraste con
lo que ocurre en el caso análogo de Schrödinger, se sabe muy poco de la estabilidad de estas soluciones.
En esta charla presentaré resultados con respecto a la estabilidad espectral de estas ondas estacionarias. 
2022-06-02
14:30hrs.
Leonardo Gordillo. Usach
Universalidad de arrugas en tubos recubiertos internamente
Edificio Felipe Villanueva
Abstract:
Cuando se somete un tubo con un recubrimiento interno a un presión negativa en su interior, se produces arrugas en su cara interna. Para estudiar este fenómeno, derivamos una ecuación simple que modela las propiedades elásticas del recubrimiento, las fuerzas de ligadura internas del recubrimiento, las fuerzas externas del sustrato y la presión impuesta. Luego, usamos continuación numérica para construir diagramas de bifurcación de las deformación en el plano en función de la presión, cuyos resultados están en excelente acuerdo con una teoría no lineal que hemos desarrollado. Este marco explica cómo la amplitud y la longitud de onda de las arrugas son seleccionadas en función de los parámetros del sistema. Más aún demostramos que la forma de las arrugas es universal en una amplia familia de sistemas de este tipo. También mostramos la aparición de pliegues y de modos mixtos. El sistema bajo análisis es el punto de inicio para comprender la formación de arrugas en los endotelios arteriales bajo cambios de presión, que han mostrado tener un rol vital en las propiedades auto-limpiantes del sistema circulatorio.

2022-05-12
14:30hrs.
Rafael Benguria . Pontificia Universidad Católica - Facultad de Física
Bounds on the maximum ionization of atoms
Edificio 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 $\mathbb{R}^3$. 
This is joint work with Juan Manuel Gonzalez and Trinidad Tubino.

https://zoom.us/j/91407685896
2021-12-01
15:45 hrs.
Danilo Polo. Pontificia Universidad Católica de Chile
Linear response theory for quantum spin systems
zoom
Abstract:
Linear response theory is a tool with which one can study systems that are driven out of equilibrium by external perturbations. The aim of this talk is to discuss the validity of linear response theory for quantum spin systems where the manifold of ground states is separated from the rest of the spectrum by a spectral gap. In addition, as one important application of the linear response theory, I provide a rigorous proof of the quantization of the orbital polarization induced by an adiabatic change of the Hamiltonian of an interacting system in a ground state, with error estimates uniform in the system size.

Zoom: https://zoom.us/j/96349024419?pwd=ckhXL2U4K2FIb3dpZUYvSmRuN1ZRZz09

Meeting ID: 963 4902 4419

Passcode: FisMat

2021-07-07
15:45 hrs.
Jaime Gómez. Pontificia Universidad Católica de Chile
Scattering Para Operadores Unitarios, Parte 2
zoom
Abstract:
En esta charla se hará una presentación de [1], en donde se exhibe una construcción de la teoría de scattering para operadores unitarios haciendo uso de los operadores de onda estacionarios. Se darán las definiciones, fórmulas y propiedades de estos operadores, asimismo de los operadores de onda fuertes y de la matriz S; además de entregar condiciones sobre los operadores de onda de modo que existan y sean iguales. Adicionalmente, se presenta un ejemplo en el cual se ilustran fórmulas de representación para los operadores de onda como para su matriz S.

[1] R. Tiedra de Aldecoa, Stationary scattering theory for unitary operators with an application to quantum walks, J. Funct. Anal. 279 (2020), no.7, 108704, 33 pp.
2021-06-30
15:45 hrs.
Jaime Gómez. Pontificia Universidad Católica de Chile
Scattering para operadores unitarios, parte 1
zoom
Abstract:
TBA
2021-03-31
15:45 hrs.
Walter Alberto de Siqueira Pedra. Universidade de São Paulo
Classical Dynamics from Self-Consistency in Quantum Mechanics
zoom
Abstract:
In 1973, Lieb and Hepp showed in a pioneering work that the time evolution of some extensive quantities related to a model for quantum spins with mean-field interaction are exactly described by classical equations of motion, at infinite volume. In particular, a Poisson bracket naturally appears in this context. During the last three decades, Pavel Bóna has developed a so-called non-linear generalization of quantum mechanics, which is based on symplectic structures for density matrices. This generalization furnishes a general abstract setting which is very convenient to study the emergence of macroscopic classical dynamics from microscopic quantum processes. In this talk, I will present a new mathematical approach to Bona's non-linear quantum mechanics, which was recently proposed in a joint work with Jean-Bernard Bru. It has a domain of applicability that is much broader than Bona's original one, and it highlights the central role of self-consistency. In this new approach, we construct a dense Poisson algebra of (weak*-)continuous functions whose domain is the set of states of any given separable unital C*-algebra. Thus, any fixed element of the Poisson algebra (which is seen as a classical Hamiltonian) yields a densely defined (generally unbounded) symmetric derivation for continuous functions. We show, by applying a notion of self-consistency, that the closure of these derivations generate positivity preserving C0-groups (i.e., Feller groups) acting on the continuous functions on the set of quantum states. In many cases, such groups preserve non-trivial Poisson subalgebras and define on them a dynamics which is equivalent to that of some simple classical mechanical system (like a classical rotor, and others). One important outcome of our method is a proof that the infinite volume dynamics of a general quantum lattice model with mean-field (long-range) interactions is always governed by an equation of motion associated with an effective short-range quantum interaction coupled to a background satisfying a classical evolution equation (Liouville's equation).
2020-12-02
15:45 hrs.
Rafael Benguria. Pontificia Universidad Católica de Chile
Estimaciones óptimas del primer autovalor del operador de Dirac en dominios suaves en dos dimensiones en términos de la geometría del dominio
https://zoom.us/j/93451228774?pwd=eUwxQzN0S1pHYXZFL0diZ1QyZWZKdz09
Abstract:
En esta charla presentaré nuevos resultados sobre la estimación del primer autovalor del operador de Dirac en dominios suaves en dos dimensiones, en términos de la geometría del dominio. Las condiciones de borde que consideramos son las llamadas "condiciones de borde masa infinita". Las estimaciones por arriba son óptimas y las por abajo casi óptimas. Este problema surge del estudio de los "puntos cuánticos". Esta charla está basada en trabajo conjunto con P. Antunes (Lisboa), V. Lotoreichik (Praga) y T. Ourmieres-Bonafos (Marsella).
2020-11-25
12:00 hrs.
Hermann Schulz-Baldes. University of Erlangen-Nuremberg
Flat bands of surface states via index theory of Toeplitz operators with Besov symbols
https://zoom.us/j/91999243096?pwd=dW5iQjVFMTh4RlR4N09OY2FNQS9Zdz09
Abstract:
The scope of the index-theoretic approach to the bulk-boundary correspondence is extended to a pseudo-gap regime. For the case of a half-space graphene model with an edge of arbitrary cutting angle, this allows to express the density of surface as a linear combination of the winding numbers of the bulk. The new technical element is an index theorem for Toeplitz operators with non-commutative symbols from a  Besov space for operators in a finite von Neumann algebra equipped with an R-action. For such operators a type II1 analogue of Peller's traceclass characterization for Toeplitz operators is proved. This is joint work with Tom Stoiber.
2020-10-28
15:45 hrs.
Marouane Assal. Pontificia Universidad Católica de Chile
Bohr-Sommerfeld quantization conditions and eigenvalue splitting for a system of coupled Schrödinger operators in the semiclassical limit
https://zoom.us/j/94064012826?pwd=Tk1QTlBUUnpBQWRWVmtCbC9FKy9lZz09
Abstract:
I will present recent results in collaboration with Setsuro Fujiié (Ritsumeikan University, Kyoto) on the distribution of eigenvalues of a one-dimensional system of coupled Schrödinger operators in the semiclassical limit. We are interested in a model where the diagonal elements of the system are Schrödinger operators on the real line with real potentials each of them admits a simple well, and the anti-diagonal elements are first order differential operators which play the role of the interaction. Such systems arise as important models in the Born-Oppenheimer approximation of molecular dynamics. We provide Bohr-Sommerfeld quantization conditions for the eigenvalues of the system in both cases where the characteristic sets admit a tangential and transversal crossings in the phase space and give precise estimates on the location of the spectrum in the semiclassical limit. In particular, in the case of symmetric wells, the eigenvalue splitting occurs and we prove that the splitting is of polynomial order, of order $h^{4/3}$ in the tangential case and of order $h^{3/2}$ in the transversal case. If I have time I will also discuss some recent results on quantum resonances for similar systems.
2020-10-21
15:45 hrs.
Vicente Lenz. Pontificia Universidad Católica de Chile
Spectral Theory for the Thermal Hamiltonian
https://zoom.us/j/95875051216?pwd=eTFMUXNiVGRlaW1zWVRLb3VsWlhSZz09
Abstract:
We will study the operator Hr, called the Thermal Hamiltonian, originally pro­posed by Luttinger to study the effects of a thermal gradient in the matter. We will start by rigurously defining the initially formally self-adjoint operator Hr, as well as sorne unitarily equivalent operators. Then we will study their spectral prop­erties, and compute their unperturbed time evolution, as their Green functions and resolvent family. We will conclude that section by presenting a convolution poten­tials family for which the scattering conditions are satisfied. Finally we will study the dynamics defined by the classical case.
2020-10-14
15:45 hrs.
Rafael Tiedra de Aldecoa. Pontificia Universidad Católica de Chile
Stationary scattering theory for unitary operators with an application to quantum walks
Zoom Meeting ID: 99457773595, Password: FisMat
Abstract:

We present a general account on the stationary scattering theory for unitary operators in a two-Hilbert spaces setting. For unitary operators $U_0,U$ in Hilbert spaces ${\mathcal H}_0,{\mathcal H}$ and an identification operator $J:{\mathcal H}_0\to{\mathcal H}$, we give the definitions and collect properties of the stationary wave operators, the strong wave operators, the scattering operator and the scattering matrix for the triple $(U,U_0,J)$. In particular, we exhibit conditions under which the stationary wave operators and the strong wave operators exist and coincide, and we derive representation formulas for the stationary wave operators and the scattering matrix. As an application, we show that these representation formulas are satisfied for a class of anisotropic quantum walks recently introduced in the literature.


https://zoom.us/j/99457773595?pwd=VXdKWEM0OUE2VkY2YWZySkRMWkxOUT09