FACULTAD DE MATEMÁTICAS

 

Seminar on Partial Differential Equations and Spectral Theory

Pontificia Universidad Católica de Chile, Campus San Joaquín 
Vicuña Mackenna 4860, Facultad de Matemáticas, Sala 2
Thursday, 17:00 - 18:30 
 
 
July 5, 2012: Low energy spectral and scattering theory for relativistic Schroedinger operators
Serge Richard, University of Tsukuba, Japan; University of Lyon I, France
 
June 7, 2012: Saint Venant Principle for Variational Inequalities
Michel Chipot, University of Zurich, Switzerland
 
April 19, 2012: Twisting versus bending in curved quantum waveguides
David Krejcirik, Nuclear Physics Institute, Czech Academy of Sciences

Abstract:

We make an overview of spectral-geometric effects of twisting and bending in quantum waveguides modelled by the Dirichlet Laplacian 
in unbounded three-dimensional curved tubes of uniform cross-section. We focus on the existence of curvature-induced eigenvalues in 
bent tubes and Hardy-type inequalities in twisted tubes of non-circular cross-section. 
Consequences of the results on the large time behaviour of the heat semigroup are also discussed.
 
 
March 22, 2012: Large time behaviour of heat kernels
Daniel Lenz, Friedrich-Schiller-Universität Jena, Germany

Abstract:

We study long time behaviour of heat kernels and show convergence of the semigroup to the ground state and convergence 
of suitably averaged logarithms of kernels to the ground state energy. The results hold for arbitrary selfadjoint positivity 
improving semigroups. This framework includes Laplace operator on manifolds, on graphs and on quantum graphs. 
(Joint work with Matthias Keller, Hendrik Vogt, Radoslaw Wojciechowski)
 
 
November 24, 2011: Positive Quantization in the Presence of a Variable Magnetic Field
Serge Richard, University of Tsukuba, Japan

Abstract:

During this seminar, we shall first recall some known results on the magnetic Weyl calculus. Then, based on a family 
of magnetic coherent states, we shall introduce a Berezin quantization for a particle in a variable magnetic field and 
we show that it constitutes a strict quantization of a natural Poisson algebra.  
 
November 17, 2011: Comportamiento local y global de las soluciones de la ecuación de Hamilton-Jacobi con operador de difusión no lineal
Amal Attouchi, Universidad de Paris 13, Francia
 
 
October 27, 2011: Fórmula de traza para el operador de Landau perturbado 
Georgi Raikov, Facultad de Matemáticas, PUC 
Resumen:
Se presentará una fórmula de traza para los cúmulos de valores propios del operador de Landau, es decir 
el operador bidimensional de Schrödinger con campo magnético constante, perturbado por un potencial 
eléctrico que decae al infinito. El espectro de este operador consiste en cúmulos de valores propios discretos 
que se acumulan en los niveles de Landau. Se estudiará la densidad asintótica de los valores propios 
dentro de estos cúmulos cuando el número del cúmulo tiende hacia el infinito. Esta densidad asintótica 
será descrita explícitamente en términos de la transformada de Radon del potencial perturbativo. Las 
herramientas usadas en las demostraciones son métodos relativos a la teoría de operadores de Berezin-Toeplitz 
y operadores pseudodiferenciales con símbolos contravariantes. Se trata de trabajo conjunto con Alexander 
Pushnitski (King's College, Londres, Reino Unido) y Carlos Villegas-Blas (UNAM, Cuernavaca, México).
Preprint
 
 
October 6, 2011: La energía de Ginzburg Landau de una función bien localizada
Alberto Montero, Facultad de Matemáticas, PUC
Resumen:
En esta charla voy a hablar de la energía de Ginzburg-Landau de una función definida en un dominio suave en R3, 
a valores complejos, cuyos conjuntos de nivel están cerca en promedio de una curva dada.
 
 
September 15, 2011: Discrete spectrum asymptotics for certain finite band lattice operators 
Alexander Sobolev, University College, London, UK

Abstract:

Discrete spectrum asymptotics have been extensively studied for classical Jacobi matrices with the dominating growing diagonal part. 
We obtain analogous asymptotic formulas for multidimensional finite band lattice operators.
The central idea of the method goes back to the Near Diagonalization Approach by G. Rozenblum, but the multi-dimensional nature 
of the problem leads to the occurrence of “resonant zones” in the lattice, which make the asymptotic properties of these operators more 
involved than in the one-dimensional case. The accurate description of the resonant zones is the main technical difficulty of this work.
 
 
September 1, 2011: Two-Hilbert spaces Mourre theory for the Laplace-Beltrami operator on manifolds 
with asymptotically cylindrical ends
Rafael Tiedra de Aldecoa, Facultad de Matemáticas, PUC

Abstract:

We review some aspects of Mourre theory in a two-Hilbert spaces setting. Then we apply this theory to the spectral analysis 
for the Laplace-Beltrami operator on manifolds with asymptotically cylindrical ends. 
This is a joint work with Serge Richard (University of Tsukuba).
 
 
July 28, 2011: Normal Form for some time-independent magnetic Hamiltonians 

Cedric Meresse, CPT Marseille, France

Abstract:

In this talk, we will discuss the existence of a normal form for some time independent quantum magnetic systems. First, we look at systems which 
have quadratic perturbations. Then, we will focus our interest on more general systems using a partial diagonalization algorithm.
 
 
May 26, 2011: El Ansatz de Bethe en una alcoba de Weyl
Erdal Emsiz, Facultad de Matemáticas, PUC

 

Resumen:

En esta charla voy a hablar sobre sistemas de raíces (de las álgebras de Lie semi-simples complejas)  versiones de partículas cuánticas en una dimensión en el círculo con potencial de delta repulsivo. Una parte importante de esta charla se dedicará al describir  cómo resolver  el problema espectral con el Ansatz de Bethe. Además, las funciones propias de Bethe son completas, en el sentido de que su span  lineal es denso en el espacio de Hilbert de las funciones cuadráticas en una alcoba de Weyl. También hablaré sobre una fórmula (conjetural) determinante  compacta para las normas cuadráticas  de las funciones propias de Bethe. Si hay tiempo también hablaré sobre la relación con álgebras de Hecke afines de dichos sistemas.

 
 
May 19, 2011: Effective Hamiltonian for a hydrogen-like atom in a thin layer
Matej Tusek, Facultad de Física, PUC

Abstract:

A hydrogen atom localized in a plane-parallel slab of the width $a$ is considered.

The energy spectrum of such atom is investigated as the width of the slab tends to zero.

It turns out that it is well approximated by the spectrum of a two-dimensional Hamiltonian which we call the effective Hamiltonian.

The norm resolvent limit of the effective Hamiltonian as $a\to 0$ is nothing but the Hamiltonian of the two-dimensional hydrogen atom plus the energy of the lowest transverse mode. Consequently, one may use the explicit knowledge of the eigenvalues of the latter Hamiltonian to approximate the eigenvalues of the exact one.

 
 
May 12, 2011: Heat kernels of two-dimensional magnetic Schroedinger and Pauli operators

Hynek Kovarik, Politecnico di Torino, Italia

Abstract:

We study the heat semi-groups generated by two-dimensional magnetic Schroedinger and Pauli operators with compactly supported magnetic field. We show that, although the spectra of these operators coincide, the respective heat kernels have different large time behavior. It turns out that the magnetic field speeds up the time decay of the Schroedinger heat kernel and slows down the decay of the Pauli heat kernel. The decay rate is in both cases determined by the total flux of the magnetic field.

 

 

April 7, 2011: Cotas para la energía de intercambio en física atómica

Rafael Benguria, PUC

Abstract:

En esta charla voy a mostrar una nueva  estimación para la energía de intercambio en Física Atómica en térmnos de un funcional que depende de la densidad de electrones. En la Teoría de Funcionales de Densidad que se emplea a menudo en Química Teórica, se trata de expresar los distintos términos de la energía de un sistema de N electrones, (los que en principio están modelados por la función de onda, solución de una ecuación de Schrödinger del sistema) por un funcional de la densidad de electrones. Lo que haré es derivar algunas nuevas desigualdades funcionales, para lograr cotas rigurosas para la energía de intercambio.  Este es un trabajo conjunto con Gonzalo Bley (PUC, Santiago) y Michael Loss (Georgia Tech, Atlanta).

 

 
January 13, 2011: Remarks on effective dynamics in Quantum Hall systems and the Chalker Coddington model
Joachim Asch, CPT Marseille, France

 

Abstract:

We consider the quantum dynamics of a single particle in the plane under the influence of a constant perpendicular magnetic and a crossed electric potential field. For a class of smooth and small potentials we prove that the Hamiltonian is unitarily equivalent to an effective Hamiltonian which commutes with the observable of kinetic energy. Further we discuss the derived quantum network percolation model suggested by Chalker and Coddington. For the restriction to a cylinder of perimeter 2M we prove simplicity of the Lyapunov exponents, finiteness of the  localization length  and compute the mean Lyapunov exponent by a Thouless formula.

 
 
January 6, 2011: Characteristic values for a class of meromorphic operators
Vincent Bruneau, Université Bordeaux I, France

 

Abstract:

Our problem (motivated by the study of resonances for some magnetic Schrödinger operators) is to study the counting function of the characteristic values for a class of meromorphic Fredohlm operators A(z). We are particularly interested by some accumulation phenomena near the singularities of A (for the study of resonances of the magnetic Schrodinger operator these singularities are the Landau levels). We will give a framework where the distribution of the characteristic values is related to the counting function of the eigenvalues of the main contribution of A.

 
 

Past Seminars