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
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
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
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
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 Levinsons
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 Berezin Toeplitz 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