Seminario de Teoría Espectral
2013-12-05
María Cristina Depassier. Facultad de Física, PUC
Pushed fronts with a cut-off: coupling the boundary layer to a variational principle
Sala 1, Facultad de Matemáticas - 17:00 Hrs.
Abstract:
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 geometr
Pushed fronts with a cut-off: coupling the boundary layer to a variational principle
Sala 1, Facultad de Matemáticas - 17:00 Hrs.
Abstract:
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 geometr
2013-11-28
David Krejcirik. Nuclear Physics Institute, Czech Academy of Sciences
The improved decay rate for the heat semigroup with local magnetic field in the plane
Sala 1 - Facultad de Matemáticas - 17:0 Hrs.
Abstract:
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 magnet
The improved decay rate for the heat semigroup with local magnetic field in the plane
Sala 1 - Facultad de Matemáticas - 17:0 Hrs.
Abstract:
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 magnet
2013-11-21
Hanne Van Den Bosch. Catholic University of Louvain y Facultad de Física, PUC
Phase transitions in PCA and associated mean field models
Sala 1 Facultad de Matemáticas - 17:00 Hrs.
Abstract:
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 t
Phase transitions in PCA and associated mean field models
Sala 1 Facultad de Matemáticas - 17:00 Hrs.
Abstract:
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 t
2013-11-14
Rafael del Rio. Universidad Nacional Autónoma de México
Spectra of Random Operators with absolutely continuous Integrated Density of States
Sala 1 Facultad de Matemáticas - 17:00 Hrs.
Abstract:
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.
Spectra of Random Operators with absolutely continuous Integrated Density of States
Sala 1 Facultad de Matemáticas - 17:00 Hrs.
Abstract:
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.
2013-11-14
Rafael del Rio. Universidad Nacional Autónoma de México
Spectra of Random Operators with absolutely continuous Integrated Density of States
Sala 1 Facultad de Matemáticas - 17:00 Hrs.
Abstract:
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.
Spectra of Random Operators with absolutely continuous Integrated Density of States
Sala 1 Facultad de Matemáticas - 17:00 Hrs.
Abstract:
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.
2013-09-26
Rafael Tiedra. Pontificia Universidad Católica de Chile
Commutator Methods for The Spectral Analysis of Time Changes of Horocycle Flows
Sala 1 Facultad de Matemáticas - 17:00 hrs.
Commutator Methods for The Spectral Analysis of Time Changes of Horocycle Flows
Sala 1 Facultad de Matemáticas - 17:00 hrs.
2013-08-29
Rafael Tiedra de Aldecoa. Pontificia Universidad Católica de Chile
The Absolute Continuous Spectrum of Skew Products of Compact Lie Groups
Sala 1 de la Facultad de Matemáticas - 17:00 Hrs.
The Absolute Continuous Spectrum of Skew Products of Compact Lie Groups
Sala 1 de la Facultad de Matemáticas - 17:00 Hrs.
2013-06-13
Serge Richard. Université Claude Bernard Lyon I
Index theorems in scattering theory: a first step towards crystals
Sala 1 - Facultad de Matemáticas - 17:00 Hrs.
Abstract:
Resumen:
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.
Index theorems in scattering theory: a first step towards crystals
Sala 1 - Facultad de Matemáticas - 17:00 Hrs.
Abstract:
Resumen:
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.
2013-04-11
Jan Felipe Van Diejen. Pontificia Universidad Católica de Chile
El espectro y scattering de un sistema de q-bosones
Sala 1 Facultad de Matemáticas - 17:00 hrs.
Abstract:
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.
El espectro y scattering de un sistema de q-bosones
Sala 1 Facultad de Matemáticas - 17:00 hrs.
Abstract:
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.
2013-04-04
Daniel Parra. Facultad de Ciencias, Universidad de Chile
Compactness Criteria for Sets and Operators in Banach Spaces
Sala 1 - 17:00 hrs. Facultad de Matemáticas - PUC
Compactness Criteria for Sets and Operators in Banach Spaces
Sala 1 - 17:00 hrs. Facultad de Matemáticas - PUC
2013-03-14
Pablo Miranda. Facultad de Física, PUC
Comportamiento asintótico de los valores propios de un Hamiltoniano magnético en el semiplano bajo condiciones de Dirichlet y Neumman.
Sala 1 de la Facultad de Matemáticas:
Abstract:
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.
Comportamiento asintótico de los valores propios de un Hamiltoniano magnético en el semiplano bajo condiciones de Dirichlet y Neumman.
Sala 1 de la Facultad de Matemáticas:
Abstract:
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.
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