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 interactions
Ivan 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, Kings 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 operator
Nicolas 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

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 Paynes 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 Levinsons theorem for this model.