Seminario de Teoría Espectral y Teoría de Dispersión
2010-05-06
Serge Richard. University of Cambridge, on Leave of Absence From The University of Lyon I
Wave operators and a topological version of Levinson´s theorem
Sala 1 -Facultad de Matemáticas - 16:30 Hrs.
Abstract:
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 ho
Wave operators and a topological version of Levinson´s theorem
Sala 1 -Facultad de Matemáticas - 16:30 Hrs.
Abstract:
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 ho
2010-04-08
Alexander Pushnitski. KingS College, London
Spectral asymptotics for Toeplitz operators in Bargman spaces
Sala 1 - Facultad de Matemáticas
Abstract:
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.
Spectral asymptotics for Toeplitz operators in Bargman spaces
Sala 1 - Facultad de Matemáticas
Abstract:
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.
2010-03-25
Nicolas Dombrowski. University of Cergy-Pontoise, Currently Postdoc At PUC
On the linear response theory and the Liouville equation for magnetic random Schrödinger operator
Sala 1 - Facultad de Matemáticas
Abstract:
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.
On the linear response theory and the Liouville equation for magnetic random Schrödinger operator
Sala 1 - Facultad de Matemáticas
Abstract:
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.
2010-01-07
Michael Loss. Georgia Institute of Technology, Atlanta
Lieb-Thirring Inequalities With Improved Constants´
Sala 2 (Víctor Ochsenius) - 16:30 Hrs. Facultad de Matemáticas - UC
Lieb-Thirring Inequalities With Improved Constants´
Sala 2 (Víctor Ochsenius) - 16:30 Hrs. Facultad de Matemáticas - UC
2010-01-07
Michael Loss. Georgia Institute of Technology, Atlanta
Lieb-Thirring Inequalities With Improved Constants
Sala 2 piso 1 Facultad de Matemáticas. 16:30 hrs.
Lieb-Thirring Inequalities With Improved Constants
Sala 2 piso 1 Facultad de Matemáticas. 16:30 hrs.
2009-12-03
Mark Ashbaugh. University of Missouri, Columbia, Mo, and Currently Visiting At The Depto.de Física, PUC
The Buckling Problem and the Krein Laplacian
Sala 2 (Víctor Ochsenius) - Facultad de Matemáticas
Abstract:
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
The Buckling Problem and the Krein Laplacian
Sala 2 (Víctor Ochsenius) - Facultad de Matemáticas
Abstract:
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
2009-10-22
Christian Enache. Universitatea Ovidius, Constanta Romania.
Bounds on the product of the first two non-trivial frequences of a free membrane
Sala 2 (Víctor Ochsenius) - 16:30 Hrs. Facultad de Matemáticas - UC
Abstract:
Resumen: 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 ine
Bounds on the product of the first two non-trivial frequences of a free membrane
Sala 2 (Víctor Ochsenius) - 16:30 Hrs. Facultad de Matemáticas - UC
Abstract:
Resumen: 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 ine
2009-10-15
Christian Enache. Universitatea Ovidius, Constanta, Romania
Some fully nonlinear elliptic boundary value problems with ellipsoidal free boundaries
Sala 2 (víctor Ochsenius) - 16:30 Hrs.
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.
Some fully nonlinear elliptic boundary value problems with ellipsoidal free boundaries
Sala 2 (víctor Ochsenius) - 16:30 Hrs.
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.
2009-09-24
Max Lein. Universidad Técnica de Munich, Alemania
A dynamical approach to the B -->0 limit for Bloch electrons (joint work with G. De Nittis)
Sala 2 (Víctor Ochsenius) - Facultad de Matemáticas -PUC
Abstract:
Resumen:
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 mag
A dynamical approach to the B -->0 limit for Bloch electrons (joint work with G. De Nittis)
Sala 2 (Víctor Ochsenius) - Facultad de Matemáticas -PUC
Abstract:
Resumen:
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 mag
2009-05-07
Rafael Tiedra de Aldecoa. Pontificia Universidad Católica de Chile
New formulae for the wave operators for a rank one interaction
Sala 2 (Vìctor Ochsenius) - Facultad de Matemáticas - 16:30 Hrs.
Abstract:
New formulae for the wave operators for a rank one interaction
Sala 2 (Vìctor Ochsenius) - Facultad de Matemáticas - 16:30 Hrs.
Abstract:
http://www.mat.puc.cl/~graikov/seminario.html )"
2008-12-19
Determinant and Breit-Wigner Approximation for Schrödinger Operators. Université de Bordeaux I
Determinant and Breit-Wigner Approximation for Schrödinger Operators
Sala 2 (Vìctor Ochsenius) - Facultad de Matemáticas - 16:30 Hrs.
Determinant and Breit-Wigner Approximation for Schrödinger Operators
Sala 2 (Vìctor Ochsenius) - Facultad de Matemáticas - 16:30 Hrs.
2008-11-28
Andreas Reisenegger. Departamento de Astronomía y Astrofisica,pontificia Universidad Católica de Chile
Equilibrios hidromagnéticos en estrellas
Sala 2 (Vìctor Ochsenius) - Facultad de Matemáticas - 16:30 Hrs.
Abstract:
Equilibrios hidromagnéticos en estrellas
Sala 2 (Vìctor Ochsenius) - Facultad de Matemáticas - 16:30 Hrs.
Abstract:
2008-11-21
Marius Mantoiu. Universidad de Chile
THE MAGNETIC QUANTUM CALCULUS (a mini-course)
Sala 2 (Víctor Ochsenius) - Fac. de Matemáticas - 16:30 Hrs.
Abstract:
THE MAGNETIC QUANTUM CALCULUS (a mini-course)
Sala 2 (Víctor Ochsenius) - Fac. de Matemáticas - 16:30 Hrs.
Abstract:
"
2008-11-14
Humberto Prado. Universidad de Santiago de Chile
Ecuaciones de Orden Fraccionario en Espacios de Banach
Sala 2 (Víctor Ochsenius) - Fac. de Matemáticas - 16:30 Hrs.
Abstract:
Ecuaciones de Orden Fraccionario en Espacios de Banach
Sala 2 (Víctor Ochsenius) - Fac. de Matemáticas - 16:30 Hrs.
Abstract:
2008-11-07
Miloslav Znojil. Doppler Institute for Mathematical Physics and Applied Mathematics, and Nuclear Physics Institute Ascr, Czech Republic
Scattering theory using non-Hermitian Hamiltonians
Sala 2 (Vìctor Ochsenius) - Facultad de Matemáticas - 16:30 Hrs.
Abstract:
Abstract: The recent progress in our understanding of the models of scattering mediated by non-Hermitian Hamiltonians will be reviewed. Several related paradoxes will be recollected and two ways (nicknamed tobogganization and discretization) towards their resolution will be outlined. Main attention will be paid to the introduction and to the analysis of properties of the metric operator $Theta$ (which is usually used as a key instrument of a return to the standard theory via an ad hoc redefinition of the Hilbert space of states in the bound state context). Mutatis mutandis a new way is proposed towards the standard theory in the scattering scenario. A few exactly solvable toy models will be offered for an explicit illustration of some technicalities.""
Scattering theory using non-Hermitian Hamiltonians
Sala 2 (Vìctor Ochsenius) - Facultad de Matemáticas - 16:30 Hrs.
Abstract:
Abstract: The recent progress in our understanding of the models of scattering mediated by non-Hermitian Hamiltonians will be reviewed. Several related paradoxes will be recollected and two ways (nicknamed tobogganization and discretization) towards their resolution will be outlined. Main attention will be paid to the introduction and to the analysis of properties of the metric operator $Theta$ (which is usually used as a key instrument of a return to the standard theory via an ad hoc redefinition of the Hilbert space of states in the bound state context). Mutatis mutandis a new way is proposed towards the standard theory in the scattering scenario. A few exactly solvable toy models will be offered for an explicit illustration of some technicalities.""
2008-10-24
Rafael Benguria. P. Universidad Católica de Chile
Relación entre la variedad nula de la transformada de Fourier de la función característica de un dominio y los autovalores del la autovalores del Laplaciano
Sala 2 (Vìctor Ochsenius) - Facultad de Matemáticas - 16:30 Hrs.
Abstract:
Resumen:
En este seminario discutiré la relación entre la distancia al origen de los ceros de la transformada de Fourier de la función
característica de un dominio y los autovalores de Dirichlet y Neumann del Lapalaciano en dicho dominio.
Relación entre la variedad nula de la transformada de Fourier de la función característica de un dominio y los autovalores del la autovalores del Laplaciano
Sala 2 (Vìctor Ochsenius) - Facultad de Matemáticas - 16:30 Hrs.
Abstract:
Resumen:
En este seminario discutiré la relación entre la distancia al origen de los ceros de la transformada de Fourier de la función
característica de un dominio y los autovalores de Dirichlet y Neumann del Lapalaciano en dicho dominio.
2008-10-17
Claudio Fernández. Pontificia Universidad Católica de Chile
Non Existence of Eigensolutions for Beams and Plates
Sala 2 (Vìctor Ochsenius) - Facultad de Matemáticas - 16:30 Hrs.
Non Existence of Eigensolutions for Beams and Plates
Sala 2 (Vìctor Ochsenius) - Facultad de Matemáticas - 16:30 Hrs.
2008-09-26
Rafael Tiedra de Aldecoa. Université de Cergy-Pontoise
Time delay for dispersive quantum Hamiltonians
Sala 2 (Víctor Ochsenius) - Fac. de Matemáticas - 16:30 Hrs.
Abstract:
Time delay for dispersive quantum Hamiltonians
Sala 2 (Víctor Ochsenius) - Fac. de Matemáticas - 16:30 Hrs.
Abstract:
2008-09-12
Georgi Raikov. Pontificia Universidad Católica de Chile
Eigenvalue asymptotics in a twisted waveguide
Sala 2 (Vìctor Ochsenius) - Facultad de Matemáticas - 16:30 Hrs.
Abstract:
Eigenvalue asymptotics in a twisted waveguide
Sala 2 (Vìctor Ochsenius) - Facultad de Matemáticas - 16:30 Hrs.
Abstract:
2008-09-05
Frédéric Klopp. Universidad de Paris 13, Francia
Localization for Some Random Quantum Graphs
Sala 2 (Vìctor Ochsenius) - Facultad de Matemáticas - 16:30 Hrs.
Localization for Some Random Quantum Graphs
Sala 2 (Vìctor Ochsenius) - Facultad de Matemáticas - 16:30 Hrs.
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