Seminario de Teoría Espectral

2017-09-28
17:00hrs.
Georgi Raikov. Facultad de Mateméticas, PUC
Teoría espectral de hamiltonianos cuánticos. II
Sala 1
http://www.mat.uc.cl/~graikov/seminar.html
2017-09-21
17:00hrs.
Georgi Raikov. Facultad de Matemáticas, PUC
Teoría espectral de hamiltonianos cuánticos. I
Sala 1
Abstract:
El propósito principal del ciclo de dos charlas "Teoría espectral de hamiltonianos cuánticos" del 21 y 28 de septiembre es despertar el interés de alumnos de licenciatura hacia los problemas matemáticos que aparecen en la teoría espectral de hamiltonianos cuánticos (operadores auto-adjuntos en espacios de Hilbert).

En la primera charla se presentarán algunos hechos básicos como el teorema espectral, la idea de cuantificación y las propiedades básicas de los operadores de Schrödinger, Pauli y Dirac.

En la segunda charla se hablará de temas más avanzados como hamiltonianos fibrados (por ejemplo, operadores con coeficientes periódicos) y hamiltonianos ergódicos.

Como ya indicado, el ciclo está orientado hacia alumnos de licenciatura pero puede ser interesante también para alumnos de postgrado y postdoctorados.
2017-08-10
17:00hrs.
Soeren Fournais. Aarhus Universiy
A Hardy-Lieb-Thirring inequality for fractional Pauli operators
Sala 1
Abstract:
In this talk we will discuss recent work on Hardy-Lieb-Thirring inequalities for the Pauli operator. The classical Lieb-Thirring inequality estimates the sum of the negative eigenvalues of a Schrödinger operator
$-\Delta + V$ by an integral of a power of the potential. In $3$-dimensions, this becomes
$$\operatorname{tr}( - \Delta + V)_{-} \leq C \int (V(x))_{-}^{5/2}\,dx$$
The classical Hardy inequality states that (also in $3D$),
$$-\Delta - \frac{1}{4 |x|^2} \geq 0,$$
where the constant $\frac{1}{4}$ is the sharp constant for this bound.

It is well known, that these inequalities can be combined to yield "Hardy-Lieb-Thirring inequalities”, i.e. the Lieb-Thirring inequality above still holds (possibly with a different constant) if $V$ is replaced by $- \frac{1}{4 |x|^2} + V$ on the left side.

In this talk we will discuss similar inequalities, where the non-relativistic kinetic energy operator $-\Delta$ is replaced by a magnetic Pauli operator. In particular, we will discuss a relativistic version, where the kinetic energy is the square root of a Pauli operator, and where $\frac{1}{4 |x|^2}$ is replaced by $\frac{c_H}{|x|}$, with $c_H$ being the critical Hardy constant for the relativistic problem.

This is joint work with Gonzalo Bley.
2017-05-18
17:00hrs.
Silvius Klein . PUC Rio de Janeiro
Anderson localization for one-frequency quasi-periodic block Jacobi operators
Sala 1
Abstract:
Consider a one-frequency, quasi-periodic, block Jacobi operator, whose blocks are generic matrix-valued analytic functions. This model is a natural generalization of Schroedinger operators of this kind. It contains all finite range hopping Schroedinger operators on integer or band integer lattices.
In this talk I will discuss a recent result concerning Anderson localization for this type of operator under the assumption that the coupling constant is large enough but independent of the frequency.
2017-04-06
17:00hrs.
Sébastien Breteaux . Basque Center for Applied Mathematics
The Time-Dependent Hartree-Fock-Bogoliubov Equations for Bosons
Sala 1
Abstract:
Joint work with V. Bach, T. Chen, J. Fröhlich, and I. M. Sigal.

It was first predicted in 1925 by Einstein (generalizing a previous work of Bose) that, at very low temperatures, identical Bosons could occupy the same state. This large assembly of Bosons would then form a quantum state of the matter which could be observed at the macroscopic scale. The first experimental realisation of a gas condensate was then done in 1995 by Cornell and Wieman, and this motivated numerous works on Bose-Einstein condensation.

In particular, we are interested in the dynamics of such a condensate. To describe the dynamics of such a condensate, the first approximation is the time dependent Gross-Pitaevskii equation, or, in an other scaling, the Hartree equation. To precise this description, we derive the time-dependent Hartree-Fock-Bogoliubov equations describing the dynamics of quantum fluctuations around a Bose-Einstein condensate via quasifree reduction. We prove global well posedness for the HFB equations for sufficiently regular interaction potentials. We show that the HFB equations have a symplectic structure and a structure similar to an Hamiltonian structure, which is sufficient to prove the conservation of the energy.
2017-03-23
17:00hrs.
Rafael Tiedra de Aldecoa. Facultad de Matemáticas, PUC
Spectral analysis of quantum walks with an anisotropic coin
Sala 1
Abstract:
We perform the spectral analysis of the evolution operator U of quantum walks with an anisotropic coin, which include one-defect models, two-phase quantum walks, and topological phase quantum walks as special cases. In particular, we determine the essential spectrum of U, we show the existence of locally U-smooth operators, we prove the discreteness of the eigenvalues of U outside the thresholds, and we prove the absence of singular continuous spectrum for U. Our analysis is based on new commutator methods for unitary operators in a two-Hilbert spaces setting, which are of independent interest.

This is a joint work with Serge Richard (Nagoya University) and Akito Suzuki (Shinshu University).
2017-03-16
17:00hrs.
Hermann Schulz-Baldes. Universidad de Erlangen, Alemania
Finite volume calculation of topological invariants
Sala 1
Abstract:
Odd index pairings of K1-group elements with Fredholm modules are of relevance in index theory, differential geometry and applications such as to topological insulators. For the concrete setting of operators on a Hilbert space over a lattice, it is shown how to calculate the resulting index as the signature of a suitably constructed finite-dimensional matrix, more precisely the finite volume restriction of the so-called Bott operator. The index is also equal to the eta-invariant of the Bott operator. In presence of real symmetries, secondary $Z_2$-invariants can be obtained as the sign of the Pfaffian of the Bott operator. These results reconcile two complementary approaches to invariants in topological insulators. Joint work with Terry Loring.
2017-01-05
17:00hrs.
Jake Fillman. Virginia Tech
Ballistic propagation for limit-periodic Jacobi operators
Sala 1
Abstract:
We will talk about the propagation of wave packets in a one-dimensional medium with limit-periodic background potential. If the amplitudes of the low-frequency modes of the potential decay sufficiently rapidly, then wavepackets travel ballistically in the sense that the group velocity is injective on the domain of the position operator. Since the underlying Hamiltonian has purely absolutely continuous spectrum, this answers a special case of a general question of J. Lebowitz regarding the relationship between ac spectrum and ballistic wavepacket spreading.
2016-12-27
15:00hrs.
Michael Loss. Georgia Tech
MODELING THERMOSTATS USING MASTER EQUATIONS
Sala 1
Abstract:
In this talk we discuss results for a model of randomly colliding particles interacting with a thermal bath, i.e., a thermostat. Collisions between particles are modeled via the Kac master equation while the thermostat is seen as an infinite gas at thermal equilibrium with inverse temperature $\beta$. The evolution propagates chaos and the one particle marginal, in the limit of large systems,  satisfies an effective Boltzmann-type  equation. The system admits the canonical distribution at inverse temperature $\beta$ as the unique equilibrium state. It turns out that any initial distribution approaches the equilibrium distribution exponentially fast, both, in a proper function space as well as in relative entropy. Recent results concerning the approximation of thermostats by a large but finite heat reservoir will also be discussed. It turns out that in suitable norms the approximation can be shown to be uniformly in time, i.e., the error depends only on the size of the finite heat reservoir. This is joint work with Federico Bonetto, Hagop Tossounian and Ranjini Vaidyanathan.
2016-12-22
14:00hrs.
Abel Klein. University of California, Irvine
Eigensystem multiscale analysis for Anderson localization in energy intervals
Sala 2
Abstract:
We present an eigensystem multiscale analysis  for proving  localization (pure point spectrum with exponentially decaying eigenfunctions, dynamical localization)  for the Anderson model in an energy interval.  In particular, it yields localization for the Anderson model in a nonempty interval at the bottom of the spectrum. This eigensystem multiscale analysis in an energy interval treats all energies of the finite volume operator at the same time, establishing level spacing and  localization of  eigenfunctions with eigenvalues in the energy interval in a fixed box with high probability.  In contrast to the usual strategy, we do not study  finite volume  Green's functions.   Instead, we perform a multiscale analysis based on finite volume eigensystems (eigenvalues and eigenfunctions).   In any given scale we only have decay for eigenfunctions with eigenvalues in the energy interval, and no information about the other eigenfunctions.  For this reason,  going to a larger scale requires new arguments that were not necessary in our previous  eigensystem multiscale analysis for the Anderson model at high disorder, where in a given scale  we have decay for all eigenfunctions. (Joint work with A. Elgart)
2016-12-01
17:00hrs.
Nicolas Popoff. Universidad de Burdeos, Francia
Ground state energy of the Robin Laplacian in corner domains.
Sala 1
Abstract:
I will consider the problem of the asymptotics of the first eigenvalue for the Laplacian with Robin boundary condition, when the Dirichlet parameter gets large. I will focus on the case where the domain belongs to a general class of corner domains, and show that the asymptotics is given at first order by the minimization of a function, called "local energy", defined on the tangent geometries. A key quantity of our analysis is the infimum of the essential spectrum of the Robin Laplacian on a cone.  Then, using a multiscale analysis, we give an estimate of the remainder. I will also provide a more precise asymptotics when the domain is regular, using a semiclassical effective Hamiltonian defined on the boundary and involving the mean curvature.
http://www.mat.uc.cl/~graikov/seminar.html
2016-12-01
15:30hrs.
Radu Purice. Instutute of Mathematics, Romanian Academy of Sciences
Spectral gaps for periodic Hamiltonians in slowly varying magnetic fields.
Sala 1
Abstract:
I report on some work done in collaboration with H. Cornean and B. Helffer. We
consider a periodic Schrödinger operator in two dimensions, perturbed by a
weak magnetic field whose intensity slowly varies around a positive mean. We
show in great generality that the bottom of the spectrum of the corresponding
magnetic Schrödinger operator develops spectral islands  separated by gaps,
reminding of a Landau-level structure.
First, we construct an effective magnetic matrix which accurately describes
the low lying spectrum of the full operator. The construction of this
effective magnetic matrix does not require a gap in the spectrum of the
non-magnetic operator, only that the first and the second Bloch eigenvalues
never cross.
Second, we perform a detailed spectral analysis of the effective matrix using
a gauge-covariant magnetic pseudo-differential calculus adapted for slowly
varying magnetic fields.
http://www.mat.uc.cl/~graikov/seminar.html
2016-11-24
17:00hrs.
Werner Kirsch. Universidad de Hagen, Alemania
On the eigenvalue distribution measure for random matrices and random Schrödinger operators
Sala 1
Abstract:
We discuss classical and recent results on the distribution of eigenvalues (density of states) for random matrices and compare them to results for random Schrödinger operators.
We discuss Wigner’s semicircle law and some of its generalizations and sketch a rather elementary proof.
http://www.mat.uc.cl/~graikov/seminar.html
2016-11-17
17:00hrs.
Nicolás Espinoza. Facultad de Matemáticas, PUC
Valores propios nulos del operador de Pauli 2D de campos magnéticos casi periódicos.
Sala 1
Abstract:
Revisamos resultados anteriores del operador de Pauli 2D: el Teorema de Aharonov-Casher y resultados acerca del operador de campos magnéticos periódicos. Luego revisamos el problema para un campo magnético casi periódico y describimos el kernel del operador de Pauli en el caso de un campo magnético particular.
http://www.mat.uc.cl/~graikov/seminar.html
2016-11-10
17:00hrs.
Galina Levitina. University of New South Wales, Australia
The principal trace formula and its applications to index theory, spectral shift function and spectral flow
Sala 1
Abstract:
Let ${A(t)}_{t\in R}$ be a one parameter family of self-adjoint operators on a separable Hilbert space $H$, that converges in norm resolvent sense to the asymptotes $A_\pm$. Consider the operator $D_A=d/dt+A(t)$ on the Hilbert space $L_2(R,H)$. Without any assumption on the spectra of the operators $A_\pm$ we prove trace formula for semigroup difference of $D_A$, which was proved initially by Robbin-Salamon under the assumption of purely discrete spectra of $A_\pm$. As a consequence of this trace formula we establish the connection between spectral shift function for the pair of the asymptotes $(A_+,A_-)$, index theory for the operator $D_A$ and the spectral flow for the family ${A(t)}_{t\in R}$ applicable for differential operators in higher dimensions. This talk is the significant extension of the results presented by Prof. Fedor Sukochev on the conference 'Spectral theory and Mathematical Physics', Santiago, 2014.

http://www.mat.uc.cl/~graikov/seminar.html
2016-11-03
17:00hrs.
Simone Murro. University of Regensburg, Germany
A novel way of constructing Hadamard state in absence of symmetry
Sala 1
Abstract:
We give a functional analytic construction of algebraic states for CAR algebras on a globally hyperbolic Lorentzian manifold. We show that in Minkowski space we recover the vacuum state and when we couple the Dirac equation to a time-dependent external potential, which is smooth and decays faster than quadratically for large times, we obtain Hadamard states.

http://www.mat.uc.cl/~graikov/seminar.html
2016-09-29
17:00hrs.
Georgi Raikov. Facultad de Matemáticas, PUC
Comportamiento asintótico de los valores propios pequeños del Laplaciano de Krein perturbado
Sala 1
Abstract:

Consideraremos el Laplaciano de Krein $K$ en un dominio acotado regular, perturbado por un multiplicador real $V$, que se anula en la frontera. Suponiendo que $V$ tiene un signo definido, vamos a discutir el comportamiento asintótico de la sucesión de valores propios de $K+V$ que tiende al origen. En particular, vamos a demostrar que el Hamiltoniano efectivo que determina el término asintótico principal, es el operador armónico de Toeplitz con símbolo $V$, unitariamente equivalente a un operador pseudodiferencial en la frontera.

Se trata de un trabajo en conjunto con Vincent Bruneau (Burdeos, Francia).

http://www.mat.uc.cl/~graikov/seminar.html
2016-09-15
17:00hrs.
Pablo Miranda. Facultad de Matemáticas, PUC
Singularidades de la Función de Corrimiento Espectral para un Hamiltoniano Magnético en el Semiplano.
Sala 1
Abstract:

En esta charla consideraremos el operador de Schrödinger $H$, con campo magnético constante, definido en un semi-plano y perturbado por un potencial $V$ que decae al infinito.  Como una posible extensión del problema de conteo de valores propios discretos de $H+V$, introduciremos la Función de Corrimiento Espectral. Probaremos que esta función es acotada en conjuntos compactos que no contienen a los valores de Landau y describiremos su comportamiento asintótico en las singularidades que se presentan  en estos valores. Para los resultados mostrados se considerará la condición de borde de Dirichlet. Resultados con la condición de Neumann también será discutidos.

http://www.mat.uc.cl/~graikov/seminar.html
2016-06-22
Rafael Tiedra de Aldecoa. P. Universidad Católica de Chile
Degree, Mixing, and Absolutely Continuous Spectrum of Cocycles With Values in Compact Lie Groups (Part 2)
Sala 1 de la Facultad de Matemáticas PUC a las 17 horas
2016-06-15
Rafael Tiedra de Aldecoa. Facultad de Matemáticas, PUC
Degree, Mixing, and Absolutely Continuous Spectrum of Cocycles With Values in Compact Lie Groups
sala 1 de la Facultad de Matemáticas a las 17:00 horas