Spectral and Scattering Theory Seminar
Pontificia Universidad Católica de Chile, Campus San Joaquín
Vicuña Mackenna 4860, Facultad de Matemáticas, Sala 1
Thursday, 16:30 - 18:00
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
May 6, 2010: Wave operators and a topological version of
Levinson's theorem
Serge
Richard, University of Cambridge, on
leave of absence from the University of Lyon I
During this
seminar, we shall first recall the definitions of the main objects of
scattering theory. We shall then introduce a common version of
Levinson's theorem that appears in the literature and discuss its meaning. This
theorem establishes a relation between the number of bound states of a
quantum system and an expression in terms of the scattering operator. Its
precise form, however, depends on various conditions, such as the dimension
of space or the existence of resonances at thresholds, and also on a regularisation procedure.
We shall then propose a different approach of this result that takes care of
the corrections and of the regularisation
automatically. In particular,
we shall show how K-theory for C*-algebras leads to a topological version of
Levinson's theorem. Our approach means, above all, a change of
perspective which makes clear that Levinson's theorem is in fact an index
theorem. Various examples will be presented and the role of the wave
operators will be emphasized. Finally, we shall sketch a new formula for the
wave operators in an abstract framework.
April 8, 2010: Spectral asymptotics for Toeplitz operators in Bargman spaces.Alexander Pushnitski, Kings College, London, UK
Abstract: I will discuss the spectrum of Toeplitz operators in Bargmann spaces. The main attention will be paid to Toeplitz operators with real symbols
of compact support. Such operators are
compact; it turns out that the eigenvalues of such operators accumulate to zero
super-exponentially fast. Moreover,
the rate of accumulation of eigenvalues
in this situation depends on the relative location of the supports of the
positive and negative parts of the symbol.
The talk is based on a recent work
with Grigori Rozenblum.
March 25, 2010: On the linear response theory and the Liouville equation for magnetic random Schrödinger operatorNicolas Dombrowski, University of Cergy-Pontoise; currently postdoc at PUC
Abstract: We consider an ergodic Schrödinger operator with magnetic field within the
non-interacting particle approximation. Justifying the linear response theory,
a rigorous derivation of a Kubo
formula for the electric conductivity
tensor within this context can be found in a
recent work of Bouclet, Germinet,
Klein and
Schenker [BoGKS].
If the Fermi
level falls into a region of localization, the well-known Kubo-Streda formula
for the quantum Hall conductivity at zero
temperature is recovered. In this review we go
along the lines of [BoGKS] but make a more systematic
use of noncommutative Lp-spaces,
leading to a somewhat
more transparent proof.
January 7, 2010: Lieb-Thirring inequalities with improved
constants
Michael Loss, Georgia Institute
of Technology, Atlanta, USA
December 10, 2009: An
Operator Algebra Approach to the Study of the Essential Spectrum of Anisotropic Quantum
Hamiltonians
Radu Purice, Romanian Academy
of Sciences
Abstract: I shall report on
two mathematical formalisms that proved to be of interest in the description of
large classes of quantum Hamiltonian systems. First on an
algebraic technique in the spectral analysis of anisotropic quantum systems.
Secondly on a symbolic calculus involving the algebra of observables and
allowing for a completely gauge invariant treatment of problems with magnetic
fields. These two methods can be put together to obtain a theorem concerning
the structure of the essential spectrum of anisotropic Hamiltonian systems in
magnetic fields. I shall present some results of this type obtained in
collaboration with W. Amrein, M. Mantoiu
and S. Richard concerning the structure of the spectrum of quantum Hamiltonians
with anizotropic potentials and magnetic fields,
supposed to be bounded and smooth. Our main result is that the essential
spectrum is the union of the spectra of the asymptotic Hamiltonians at infinity
defined in terms of the compactification of the usual
n-dimensional space associated to the given class of potentials and magnetic
fields.
December 3, 2009: The
Buckling Problem and the Krein Laplacian
Mark Ashbaugh, University of
Missouri, Columbia, MO, currently visiting at the Physics Faculty, PUC
Abstract: Recent
developments on the buckling problem and the Krein Laplacian in which the author has been
involved will be discussed, including connections between these
two problems, analysis of their spectral asymptotics,
and inequalities for their eigenvalues. In particular, we show how the buckling problem is intimately related to the Krein
Laplacian, and that, in fact, there is a unitary equivalence
between the two problems if one considers the Krein Laplacian on the space orthogonal (in
an appropriate sense) to its kernel.
Old conjectures
concerning the eigenvalues of
the buckling problem will also be presented, including the Polya-Szego conjecture for the first eigenvalue (which would be the Faber-Krahn result for this problem) and
Paynes conjecture comparing the buckling eigenvalues to those of the Dirichlet Laplacian on the same domain.
The Krein
Laplacian
was
first discussed by von Neumann around 1930 in the context of extensions of
operators, though perhaps without a full understanding of its significance. Later, in
two long papers in 1947, Krein made an extensive
investigation of it, presenting its most important properties and elucidating its place among the possible self-adjoint extensions
of the Laplacian. It turns out that the Krein-von Neumann extension of the Laplacian (which we refer to briefly as the Krein Laplacian) is the minimal
nonnegative self-adjoint extension of the Laplacian. This is to be contrasted with
the fact that the Friedrichs extension is the maximal such extension. Because of
this, the Krein-von Neumann extension is often
referred to as the ``soft extension, while the Friedrichs extension is referred to as the ``hard extension.
Much of the recent work presented in the talk represents joint work with
Fritz Gesztesy, Marius Mitrea,
Roman Shterenberg, and/or Gerald Teschl.
October 22, 2009: Bounds
on the product of the first two non-trivial frequences of a free membrane.
Christian Enache, Ovidius University,
Constanta, Romania
Abstract: In this talk we
are interested in the eigenvalue problem of a free membrane represented as a
bounded simply-connected planar domain D with Lipschitz
boundary. The aim of this talk is twofold. First, we give a positive answer to
the following conjecture of Iosif Polterovich:
the product of the first two non-trivial Neumann eigenvalues of the laplacian on D (frequencies of the free membrane D) is upper bounded by the value of the same quantity for the disk
with same area as D. This estimate is sharp and the equality occurs if and only
if D is a disk. Secondly, we consider the class of n-sided convex polygons and
establish an isoperimetric inequality for the product of some moments of
inertia. As an application, we obtain an explicit nice upper bound for the
product of the first two non-trivial frequences of a
free membrane represented as a n-sided convex polygon.
October 15, 2009: Some
fully nonlinear elliptic boundary value problems with ellipsoidal free
boundaries
Christian Enache, Ovidius University, Constanta, Romania
Abstract: In this talk we will present some overdetermined boundary value problems for three classes of fully nonlinear elliptic
equations. In each case we prove that the solution exists if and only if the underlying domain is the interior of an ellipsoid
(or ellipse in two dimensions). The proofs make use of the uniqueness theorem, maximum principles or some geometric arguments
involving the curvatures of the free boundary.
September 24, 2009: A dynamical approach to the B --> 0 limit for Bloch electrons
Max Lein, Technical
University, Munich, Germany
(joint work
with G. De Nittis)
Abstract: Consider a particle
in a periodic potential subjected to an external electromagnetic field. If some
of the Bloch bands are separated from the others by a gap, transitions are
exponentially suppressed and we can write down effective dynamics for initial
states in this band. This problem naturally has a multiscale
formulation with three scales: microscopic vs. macroscopic spatial scale \varepsilon and ratio between electric and magnetic field
\lambda.
To leading order in
\varepsilon \ll 1, the
dynamics are still generated by the Bloch bands even if a constant magnetic
field is present. The first- order correction is essentially given by B \cdot \mathcal{M} where \mathcal{M} is an effective magnetization of the relevant
Bloch bands (the so called Rammal-Wilkinson term). Hence, in terms of dynamics,
we recover the Bloch band picture even if B = const. In particular, our
results show that the B --> 0 limit is well-behaved in the sense of dynamics. Our results
hold for magnetic fields whose components are bounded with bounded derivatives to any order.
In addition, we are
able to approximate the dynamics semiclassically in a
separate step via an Egorov-type theorem.
May 7, 2009: New
formulae for the wave operators for a rank one interaction
Rafael Tiedra
de Aldecoa, Pontificia Universidad Católica de Chile
Abstract: We prove new
formulae for the wave operators for a Friedrichs
scattering system with a rank one perturbation, and we derive atopological version
of Levinsons theorem for this model.