FACULTAD DE MATEMÁTICAS
Seminar on Partial Differential
Equations and Spectral Theory
(former
Spectral and Scattering Theory Seminar)
Pontificia Universidad Católica de Chile, Campus San Joaquín Vicuña Mackenna 4860, Facultad de Matemáticas, Sala 2Thursday, 17:00 - 18:30 Contact: bourget(at)mat.puc.cl; mcourdurier(at)mat.puc.cl, amontero(at)mat.puc.cl; graikov(at)mat.puc.cl November 24, 2011: Positive Quantization in the Presence of a Variable Magnetic FieldSerge 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, PUCResumen: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.
December 9, 2010: Universally Typical Sets for Ergodic Sources of Multidimensional Data
Ruedi Seiler, Technische Universität Berlin, Germany
Abstract:
Results about typical sets are lifted
from the one-dimensional to the multidimensional setting. This is a step
towards a multidimensional Lempel-Ziv Compression algorithm.
November 11, 2010: On resonance and the Landerman-Lazer condition for some Fully Nonlinear Elliptic equations Alexander Quaas, Universidad Téctinca Federico Santa María
October 28, 2010: Large Solutions of Some Elliptic Systems of
Second Order
Marta García-Huidobro, PUC
Abstract:
We study the nonnegative solutions of
the elliptic system \[\Delta u=|x|^{a}v^{\delta},\qquad\Delta
v=|x|^{b}u^{\mu}%\] in the superlinear case
$\mu\delta>1,$ which blow up near the boundary of a ball. In the
radial case we give the exact behavior of the large solutions near the
boundary.
October 21, 2010: The system of elastic waves with time-dependent damping coefficient in unbounded domains
Ruy
Coimbra, Universidade
Federal de Santa Catarina
October 4, 2010 (Monday): The Relativistic Scott Correction
Heinz Siedentop, Ludwig-Maximilians-Universität München, Germany
September 30, 2010: Spectral properties of random Schroedinger operators with general interactionsIvan Veselic, Chemnitz Technical University, Germany
Abstract
We discuss spectral properties of
random Schroedinger operators, in particular those related
to the phenomenon of Anderson localisation.
After introducing the most studied
model, namely the alloy or Anderson type potential, we consider several other
types of random operators with more general interactions. There are several
properties those models may exhibit or not: monotone and/or linear dependence
on the random parameters, long range correlations, randomness entering only a
multiplication operator etc.
We discuss the relevance of these
properties when deriving certain intermediate spectral estimates, and explain
new insights which are gained about the physical mechanism leading to localisation.
September 9, 2010: Ground states para una ecuación elíptica semilineal con crecimiento mixto subcrítico y supercrítico
Ignacio
Guerra, Departamento de Matemática y CC, USACH
Abstract
Consideraremos
el problema $\Delta u + u^p+\lambda u^q -u = 0$,
$u>0$, en $\mathbb{R}^N$, donde $\lambda>0$, $N\geq
3$ y $1<q\leq
p$. Haremos una descripción numérica de las soluciones para el caso $q$ subcrítico y $p$ supercrítico. Además probaremos existencia
cuando $q$ es subcrítico y $p$ es supercritico
cercano al exponente crítico. Discutiremos también la multiplicidad, unicidad y
no existencia de soluciones para este problema.
September 2, 2010: Sojourn Time for Time-Dependent Hamiltonians
Claudio Fernández, PUC
August 26, 2010: Expanding solitons with non-negative curvature operator coming out of cones.
Felix Schulze,
Free University Berlin, Germany
Abstract
We show that a Ricci flow of any complete
Riemannian manifold without boundary with bounded non-negative curvature
operator and non-zero asymptotic volume ratio exists for all time and has
constant asymptotic volume ratio. We show that there is a limit solution,
obtained by scaling down this solution at a fixed point in space, which is an
expanding soliton coming out of the asymptotic cone
at infinity.
August 19, 2010: Curvas invariantes y cociclos de isometrías
en un espacio de Hilbert sobre dinámicas minimales.
Abstract
Esta
charla trata sobre cociclos de isometrías en un
espacio de Hilbert sobre dinámicas minimales y las dinámicas fibradas
que ellos generan. Veremos que una condición necesaria para la existencia de
una curva invariante en estas dinámicas fibradas es
que el cociclo sea acotado. En dimensión finita
mostraremos que esta condición es suficiente, lo que da una version
general de un teorema de Gottschalk y Hedlund. Daremos algunos ejemplos en dimensión infinita
donde la condición de cociclo acotado implica la
existencia de una curva debilmente continua.
Finalmente mostraremos que toda curva debilmente
continua es fuertemente continua.
Este
es un trabajo conjunto con Andrés Navas y Mario Ponce.
August 12, 2010: A limiting free boundary problem ruled by Aronsson's
equation
Julio
Rossi, Universidad de Buenos Aires, Argentina
Abstract
We study the behavior of $p$-Dirichlet optimal design problem with volume constraint for
$p$ large. As the limit as $p$ goes to infinity, we find a limiting free
boundary problem governed by the infinity-Laplacian
operator. We find a necessary and sufficient condition for uniqueness of the
limiting problem and, under such a condition, we
determine precisely the optimal configuration for the limiting problem.
Finally, we establish convergence results for the free boundaries.
(joint work
with Eduardo V. Teixeira (Brasil))
August
5, 2010: Soluciones múltiples al
problema de Bahri-Coron en un dominio con una
perturbación topológicamente no trivial
Mónica Clapp, Universidad Nacional Autónoma de México