Seminario de Teoría Espectral

Pontificia Universidad Católica de Chile, Campus San Joaquín

Vicuña Mackenna 4860, Facultad de Matemáticas, Sala 1

Jueves, 17:00 - 18:30

 

 

4 de diciembre de 2014: Equality of bulk and edge Hall conductances for random magnetic Schroedinger operators

Amal Taarabt, Facultad de Física, PUC

Abstract:
We are interested in the bulk and edge Hall conductances for continuous models in the presence of magnetic or electric walls. We explain how the walls come into play in order to define the edge conductance. We shall take into account the contribution of localized states and consider a regularization that a disordered media requires. We prove the equality of these conductances by deriving one from the other, and not by separate quantization.

 

International Conference Spectral Theory and Mathematical Physics

24 - 28 de noviembre 2014

 

Escuela de Operadores de Schroedinger Aleatorios

13 - 21 de noviembre 2014

 

6 de noviembre de 2014: Commutator criteria for strong mixing

Rafael Tiedra de Aldecoa, Facultad de Matemáticas, Pontificia Universidad Católica de Chile

Abstract:

We present new criteria, based on commutator methods, for the strong mixing property of discrete flows {U^N} and continuous flows {e^(itH)} induced by unitary operators U and self-adjoint operators H in a Hilbert space \H. Our approach put into evidence a general definition for the topological degree of the curves N->U^N and t->e^(itH) in the unitary group of \H. As an example, we present an application to time changes of horocycle flows.

 

30 de octubre de 2014: Local Spectral Asymptotics for Metric Perturbations of the Landau Hamiltonian

Tomás Lungenstrass, Facultad de Matemáticas, Pontificia Universidad Católica de Chile
Abstract:

We consider metric perturbations of the Landau Hamiltonian. We investigate the asymptotic behavior of the discrete spectrum of the perturbed operator near the Landau levels, for perturbations with power-like decay, exponential decay or compact support.

This is joint work with Georgi Raikov

 

23 de octubre de 2014: Counter-examples to strong diamagnetism

Soeren Fournais, Aarhus University, Denmark
Abstract:

Consider a Schrödinger operator with magnetic field $B(x)$ in 2-dimensions. The classical diamagnetic inequality implies that the ground state energy, denoted by $\lambda_1(B)$, with magnetic field is higher than the one without magnetic field. However, comparison of the ground state energies for different non-zero magnetic fields is known to be a difficult question. We consider the special case where the magnetic field has the form $b \beta$, where $b$ is a (large) parameter and $\beta(x)$ is a fixed function. One might hope that monotonicity for large field holds, i.e. that $\lambda_1(b_1 \beta) > \lambda_1(b_2 \beta)$ if $b_1>b_2$ are sufficiently large. We will display counterexamples to this hope and discuss applications to the theory of superconductivity in the Ginzburg-Landau model.

This is joint work with Mikael Persson Sundqvist.

 

16 de octubre de 2014: Eigenvalue asymptotics for the perturbed Iwatsuka Hamiltonian

Pablo Miranda, Escuela de Ingeniería, Pontificia Universidad Católica de Chile

Abstract:

In this talk we will give the description of the discrete spectrum of a two-dimensional Schrödinger operator H with a non constant magnetic field B that depends only on one of the variables, and an electric potential V that decays at infinity.  In particular, we will consider the problem of the number of eigenvalues of H in the gaps of its essential spectrum.

First we will describe effective Hamiltonians that are valid under some general conditions, and then we will use them to find the asymptotic behavior of the eigenvalues when the potential V is a power-like decaying function and when is a compactly supported function, showing a semi-classical behavior of the eigenvalues in the first case and a non semi-classical behavior in the second one.

 

9 de octubre de 2014: Lieb-Thirring type inequalities for non self-adjoint Schrödinger operators

Diomba Sambou, Facultad de Matemáticas, Pontificia Universidad Católica de Chile

Abstract:

We present new results of Lieb-Thirring type inequalities on the discrete spectrum of (magnetic) non self-adjoint Schroedinger operators. In particular, these inequalities give a priori information on the distribution of the discrete spectrum (complex eigenvalues) near the essential spectrum, and describe how sequences of eigenvalues converge.

 

2 de octubre de 2014: Radiative corrections to the binding energy for a spin $1/2$ charged particle

Semjon Wugalter, Karlsruhe Institute of Technology, Germany

Abstract:

The talk is based on several joint works with J.-M. Barbaroux, Th. Chen and V. Vougalter. We compute the binding energy of a Hydrogen atom for two the most comprehensive models in nonrelativistic QED. From mathematical point of view the problem is interesting due to the facts that the unperturbed eigenvalue belongs to the essential spectrum of the operator, the perturbation is not analytic and not small.

 

25 de septiembre de 2014: Ergodicity and localization for the Delone-Anderson model

Constanza Rojas-Molina, LMU Munich, Germany

Abstract:

Delone-Anderson models arise in the study of wave localization in random media, where the underlying configuration of impurities in space is aperiodic, as for example, in disordered quasicrystals. The lack of translation invariance in the model yields a break of ergodicity, and the loss of properties linked to it. In this talk we will present recent results on the ergodic properties of such models, namely, the existence of the integrated density of states and the almost-sure spectrum. We use the framework of coloured Delone dynamical systems, which allows us to retrieve properties known for the ergodic Anderson model, under some geometric assumptions on the underlying configuration of impurities.  In the particular case of a Delone-Anderson perturbation of the Laplacian, we can prove that the integrated density of states exhibits a Lifshitz-tail behavior, which allows us to study localization at low energies. This is joint work with F. Germinet (U. de Cergy-Pontoise) and P. Müller (LMU Munich).

 

11 de septiembre de 2014: The magnetic Weyl calculus: a Lie theoretic point of view

Ingrid Beltita, Institute of Mathematics "Simion Stoilow" of the Romanian Academy

Abstract:

We present a Weyl calculus for pseudo-differential operators on nilpotent Lie groups that takes into account magnetic fields, not necessarily polynomial. This requires an infinite-dimensional Lie group, which is the semidirect product of a nilpotent Lie group and an appropriate function space thereon. We single out a certain finite dimensional coadjoint orbit of that semidirect product and construct our pseudo-differential calculus as a Weyl quantization of that orbit. We also discuss spaces of symbols for this Weyl calculus.

In the case when the nilpotent group is the additive group of some finite-dimensional vector space, we recover the magnetic pseudo-differential calculus constructed by V. Iftimie, M. Mantoiu and R. Purice.

The lecture reports on joint work with Daniel Beltita (IMAR, Bucharest).

 

21 de agosto de 2014: Properties of Coulombic eigenfunctions of atoms and molecules

Thomas Soerensen, LMU Munich, Germany

Abstract:

The eigenfunctions of the Schroedinger operator for (non-relativistic) atoms and molecules (in the Born-Oppenheimer/clamped nuclei approximation) are solutions of an elliptic partial differential equation with singular (total) potential (ie, zero-order term). In this talk we give an overview over our results about the structure/regularity of the eigenfunctions at the singularities of the potential. These, in particular, improve on the well-known 'Kato Cusp Condition'. If time permits, we also discuss the implications for the electron density.

This is joint work with S. Fournais (Aarhus, Denmark), and T. Hoffmann-Ostenhof (Vienna, Austria).

 

3 de julio de 2014: Asymptotic stability of soliton states of nonlinear Schrödinger equations
Manuel Larenas, Rutgers University, USA

Abstract:

The nonlinear Schrödinger equation (NLS) has in general localized solutions which are known as soliton states. If the initial data is given by a sum of solitons plus a small perturbation, under suitable conditions the solution has been proven to exhibit the asymptotic profile of independently moving solitons plus decaying radiation. The argument requires linearization of the NLS around the bulk term and to establish dispersive estimates for the linear problem. I will present different methods to find these estimates, including a new, abstract approach that extends to spectral thresholds and high energy.

 

29 de mayo de 2014: Transporte cuántico en sistemas de baja dimensionalidad

Enrique Muñoz, Instituto de Física, PUC

Resumen:
Uno de los problemas teóricos modernos en la física de la materia condensada,  motivados por la miniaturización de dispositivos microelectrónicos, es el transporte a través de sistemas cuánticos de baja dimensionalidad. El modelamiento de estos fenómenos requiere el desarrollo de teorías apropiadas para sistemas cuánticos fuera del equilibrio, donde el formalismo de Keldysh representa una alternativa muy versátil. En esta charla, se discutirán aspectos generales e introductorios del formalismo de Keldysh, y se discutirá un ejemplo de aplicación para el transporte a través de un punto cuántico, más allá del régimen de respuesta lineal.

 

15 de mayo de 2014: Global Bounds on the Period of Nonlinear Oscillators

Rafael Benguria, Facultad de Física, PUC

Abstract:

We use a variational characterization of the period of nonlinear oscillators in order to find sharp global bounds, for a general class of potentials.

This is joint work with M.C. Depassier (PUC) and M. Loss (Georgia Tech).

 

8 de mayo de 2014: On Schroedinger Operator with Quasi-periodic Potential in Dimension Two

Yulia Karpeshina, University of Alabama at Birmingham, USA

Abstract

 

10 de abril de 2014: Resolvent expansions and continuity of the scattering matrix at embedded thresholds

Rafael Tiedra de Aldecoa, Facultad de Matemáticas, PUC

Abstract:

We present an inversion formula which can be used to obtain resolvent expansions near embedded thresholds. As an application, we prove for a class of quantum waveguides the absence of accumulation of eigenvalues and the continuity of the scattering matrix at all thresholds.

 

27 de marzo de 2014: Asintóticas de Lifschitz en la red de Bethe

Francisco Hoecker, TU Chemnitz

Resumen:
La integral de la densidad de estados (o función de conteo de valores propios normalizada) asociada a modelos de medios desordenados presenta un decaimiento exponencial cerca de los bordes de las bandas espectrales. Este fenómeno es conocido como comportamiento asintótico de Lifschitz y en algunos casos indica la existencia de localización de Anderson (ausencia de difusión de los paquetes de ondas). En esta charla hablaremos sobre el decaimiento de la integral de la densidad de estados del modelo de Anderson cuyo espacio físico subyacente es un retículo de Bethe (el grafo de Cayley de un grupo libre). Esto es trabajo en conjunto con C. Schumacher.

 

9 de enero de 2014: Nonlinear flows and rigidity results on compact manifolds

Michael Loss, Georgia Tech

Abstract:

This talk is about a certain class of non-linear PDEs on a compact connected Riemannian manifolds without boundary.

The problem is to prove that there are no solutions other than the constant function.  These rigidity results yield sharp Sobolev type inequalities. While some of the results date back to the 90-ies, a new perspective has emerged in the last five years.

The idea is to use porous media or fast diffusion flows that yield relatively straightforward proofs for such rigidity results.

This is joint work with Jean Dolbeault and Maria ´Esteban.

 

19 de diciembre de 2013: Risk estimation for regularized regression problems

Carlos Sing-Long, Stanford University

Abstract:

In many problems in science and engineering one wants to recover an object from incomplete information obtained from linear measurements. In practice the measurements are corrupted by noise and therefore exact recovery is not possible. When the underlying object has some a priori known structure, a popular approach is to use a regularized maximum likelihood estimator obtained by solving a convex optimization problem. The objective function consists of two terms, one that enforces data consistency, usually the likelihood, and another that enforces the known structure in the object. Typically this trade-off is controlled by a non-negative scalar multiplying the regularizer. This procedure yields a family of estimates parametrized by the value of this scalar. Intuitively, some values will produce more accurate estimates of the true object than others. This talk addresses the problem of using unbiased estimates for the statistical risk, that is, the expected mean-squared error between the true object and the estimate, as a way to select a value. We will discuss recent advances toward a derivation of explicit expressions for such an estimator for a widely used class of regularizers. We will also explore the connections between the geometry of the problem and the risk estimate.

 

5 de diciembre de 2013: Pushed fronts with a cut-off: coupling the boundary layer to a variational principle.

María Cristina Depassier, Facultad de Física, PUC

Abstract:

We study the change in the speed of pushed and bistable reaction diffusion fronts of the reaction diffusion equation in the presence of a small cut-off. We give explicit formulas for the shift in the speed for arbitrary reaction terms $f(u)$. The dependence of the speed shift on the cut-off parameter is a function of the front speed and profile in the absence of the cut-off.  In order to determine the power law dependence of the speed shift on the cut-off parameter we solve the leading order approximation to the front profile $u(z)$ in the neigborhood of the leading edge and use a variational principle for the speed. We apply the general formula to the Nagumo equation and recover the results which have been obtained recently by geometric blow up analysis.

 

28 de noviembre de 2013: The improved decay rate for the heat semigroup with local magnetic field in the plane

David Krejcirik, Nuclear Physics Institute, Czech Academy of Sciences

Abstract:

We consider the heat equation in the presence of compactly supported magnetic field in the plane. We show that the magnetic field leads to an improvement of the decay rate of the heat semigroup by a polynomial factor with power proportional to the distance of the total magnetic flux to the discrete set of flux quanta.

The proof employs Hardy-type inequalities due to Laptev and Weidl for the two-dimensional magnetic Schroedinger operator and the method of self-similar variables and weighted Sobolev spaces for the heat equation. A careful analysis of the asymptotic behavior of the heat equation in the similarity variables shows that the magnetic field asymptotically degenerates to an Aharonov-Bohm magnetic field with the same total magnetic flux, which leads asymptotically to the gain on the polynomial decay rate in the original physical variables.

 

21 de noviembre de 2013: Phase transitions in PCA and associated mean field models

Hanne van den Bosch, Catholic University of Louvain, Belgium, and Faculty of Physics, PUC

Abstract:

Probabilistic cellular automata (PCA) are a special kind of Markov chains that are studied in mathematical physics and computer science. In these models both space and time are discretized, which allows for a simple formulation and easy numerical simulation. In spite of this apparent simplicity, PCA feature a wide variety of interesting phenomena. In particular, the competition between random noise and some deterministic transition rule may give rise to two opposed types of long term behavior: ergodicity when all information about the initial condition disappears as time tends to infinity, versus non-ergodicity when the asymptotic state depends on the initial condition. The transition between both regimes is called a dynamical phase transition. The talk will give an overview of the main mathematical results concerning phase transitions in PCA together with the intuitive idea of their proofs. These exact results will be contrasted with the ones obtained in a mean field approximation.

 

14 de noviembre de 2013: Spectra of Random Operators with absolutely continuous Integrated Density of States

Rafael del Rio, Universidad Nacional Autónoma de México

Abstract:

The talk will be about the structure of the spectrum of random operators.  Basic definitions about random operators will be reviewed and it will be show that if the density of states measure of some subsets of the spectrum is zero, then these subsets are empty. In particular it follows that absolute continuity of the IDS implies singular spectra of ergodic operators is either empty or of positive measure. Our results apply to Anderson and alloy type models, perturbed Landau Hamiltonians, almost periodic potentials and models which are not ergodic.

 

7 de noviembre de 2013: Confinement-deconfinement transitions for two-dimensional Dirac particles

Josef Mehringer, LMU Munich, Germany, and Faculty of Physics, PUC

Abstract:

We consider a two-dimensional massless Dirac-Operator H coupled to a magnetic field B and a scalar potential V growing at infinity. We describe features of the spectrum of H depending on the relation of V and B at infinity. In particular a sharp condition for the discreteness of the spectrum will be given. Beyond this condition we find dense pure point spectrum. In addition we give an outlook for applications of our techniques to two-dimensional magnetic Schroedinger-/Pauli-Operators and discuss open questions arising from our results.

 

24 de octubre de 2013: Matrix-valued orthogonal polynomials

Erik Koelink, Radboud Universiteit Nijmegen, the Netherlands

Abstract:

Matrix-valued orthogonal polynomials date back to the 50ies in the work of M.G. Krein, and have been studied recently from various points of view. After discussing some general elementary properties we discuss two set-ups that give rise to new examples and applications of matrix-valued orthogonal polynomials. The first set-up is related to spectral theory of some explicit suitable operators, and the second is related to representation theory.

 

17 de octubre de 2013: Estudio de Transformaciones de Furstenberg usando una Estimación de Mourre

Paulina Cecchi, Facultad de Ciencias, Universidad de Chile

Resumen:

Las transformaciones de Furstenberg son un tipo de skew product, objetos ampliamente estudiados en Sistemas Dinámicos. En esta charla veremos cómo se puede abordar el análisis espectral del operador de Koopman (U_T) asociado a una transformación de Furstenberg 'general' (en un espacio más general que aquel introducido por el propio Furstenberg en los '60), utilizando herramientas de la teoría de conmutadores.   Veremos que el operador U_T tiene espectro puramente absolutamente continuo restringido al subespacio de funciones que sólo dependen de la primera componente en L^2(X), donde X es el espacio sobre el cual está definida nuestra transformación.

 

26 de septiembre de 2013: Commutator methods for the spectral analysis of time changes of horocycle flows

Rafael Tiedra de Aldecoa, Pontificia Universidad Católica de Chile

Abstract:

We show that all time changes of the horocycle flow on compact surfaces of constant negative curvature have purely absolutely continuous spectrum in the orthocomplement of the constant functions. This provides an answer to a question of A. Katok and J.-P. Thouvenot on the spectral nature of time changes of horocycle flows. Our proofs rely on positive commutator methods for self-adjoint operators and the unique ergodicity of the horocycle flow.

 

29 de agosto de 2013: The absolute continuous spectrum of skew products of compact Lie groups

Rafael Tiedra de Aldecoa, Pontificia Universidad Católica de Chile

Abstract

 

22 de agosto de 2013: Existence and stability of periodic solutions for a class of differential delay equations

Anatoli F. Ivanov, Pennsylvania State University, USA

Abstract

 

11 de julio de 2013: Lower bound for the first eigenvalue of the Laplacian on manifolds with bounded Ricci curvature

Julie Clutterbuck, Australian National University, Canberra

Abstract:

We derive gradient estimates for solutions of the heat equation on a compact manifold with Ricci curvature bounded from below.  These estimates give a new and simple proof of the lower bound for the first eigenvalue on such manifolds found by Kroeger and Bakry-Qian.      

This is joint work with Ben Andrews.

 

20 de junio de 2013: Una Fórmula de Traza para Perturbaciones de Largo Alcance del Hamiltoniano de Landau

Tomás Lungenstrass, Facultad de Matemáticas, PUC

Resumen

 

13 de junio de 2013: Index theorems in scattering theory: a first step towards crystals

Serge Richard, Université Claude Bernard Lyon I

Abstract:

During this talk, we shall look at some possible extensions of the framework developed for a topological approach of Levinson’s theorem. One such extension would be to investigate crystals and their defects through scattering theory together with non commutative topology. As a first illustration of our aim, we shall recall the scattering theory for the Laplacian with a periodic boundary condition, and reinterpret this example in our setting.

 

16 de mayo de 2013: Un sistema de q-bosones  con una interacción en el borde

Erdal Emsiz, Facultad de Matemáticas, PUC

Resumen:

Los q-bosones constituyen un sistema de partículas cuánticas en el espacio de Fock caracterizado por operadores de creación y aniquilación satisfaciendo relaciones de conmutación tipo q Heisenberg. Consideramos los q-bosones en la retícula semi-infinita $\mathbb{N}$, pero modificamos en el punto final los operadores de creación y de aniquilación tal que representan una deformación cuadrática de las relaciones de conmutación  mencionada arriba.

Combinando con una perturbación diagonal llegamos al Hamiltoniano de un sistema de q-bosones  con una interacción en el borde parametrizado por 2 parámetros. Demostraremos que el Hamiltoniano tiene espectro absolutamente continuo y calculamos además el operador de scattering usando el principio de la fase estacionaria.

 

25 de abril de 2013: Resonancias y singularidades en los umbrales espectrales para hamiltonianos cuánticos magnéticos

Georgi Raikov, Facultad de Matemáticas, PUC

Resumen:

Sean  H_0 el operador de Schroedinger en tres dimensiones con campo magnético constante, V potencial eléctrico que decae suficientemente rápido en infinito, y H = H_0 + V. Primero, consideraremos el comportamiento asintótico de la función de Krein de corrimiento espectral (SSF de "spectral shift function") para el par de operadores (H, H_0) cerca de los niveles de Landau que tienen el rol de umbrales en el espectro de H_0. Mostraremos que la SSF tiene singularidades cerca de los niveles de Landau y describiremos estas singularidades en términos de ciertos operadores compactos de Berezin - Toeplitz. 

Luego, definiremos las resonancias para el operador H e investigaremos su distribución asintótica cerca de los niveles de Landau. Demostraremos que, bajo hipótesis apropiadas sobre el potencial V, existe un número infinito de resonancias cerca de cada nivel de Landau fijo. Encontraremos el término asintótico principal de la correspondiente función de conteo de resonancias que se escribe a través de los mismos operadores de BerezinToeplitz que aparecen en la descripción de las singularidades de la SSF.

La charla es basada en trabajos conjuntos con J.-F. Bony (Burdeos), V. Bruneau (Burdeos), C. Fernández (Santiago de Chile), Alexander Pushnitski (Londres) y Simone Warzel (Munich).

 

11 de abril de 2013: El espectro y scattering de un sistema de q-bosones

Jan Felipe Van Diejen, Facultad de Matemáticas, PUC

Resumen:

Los q-bosones constituyen un sistema de partículas cuánticas en el espacio de Fock caracterizado por operadores de creación y aniquilación satisfaciendo relaciones de conmutación tipo q Heisenberg. Demostraremos que el Hamiltoniano tiene espectro absolutamente continuo y calculamos el operador de scattering usando el principio de la fase estacionaria.

 

4 de abril de 2013: Compactness criteria for sets and operators in Banach spaces

Daniel Parra, Facultad de Ciencias, Universidad de Chile

Abstract:

We consider families of operators indexed by a topological space; this family allows us to characterize compact subsets of a Hilbert space. Our main result is both a generalization of Riesz-Kolmogorov theorem and also an extension of compacity results based on representation coefficients. We will then generalize part of our results in the coorbit setting.

 

14 de marzo de 2013: Comportamiento asintótico de los valores propios de un Hamiltoniano magnético en el semiplano bajo condiciones de Dirichlet y Neumman

Pablo Miranda, Facultad de Física, PUC

Resumen:

En esta charla consideramos dos operadores de Schrödinger con campo magnético constante en un semiplano, uno definido con condiciones de borde de Dirichlet y el otro con condiciones de Neumman. Si V es un potencial real no-positivo que decae al infinito, estudiamos el espectro discreto de los operadores originales perturbados por V. En el caso de Dirichlet mostramos que incluso cuando la perturbación V es muy débil, aparecerán infinitos valores propios bajo el espectro esencial del operador, mientras que en el caso de Neumann esto dependerá de la velocidad de decaimiento de V.

Este es trabajo conjunto con G. Raikov y V. Bruneau.

 

Seminarios 2010-2012; Seminarios 2008-2009