# Seminario de Teoría Espectral

2018-12-13
17:00hrs.
Ivan Veselic. Tu Dortmund
Wegner estimate for Landau-breather Hamiltonians
Sala 2
Abstract:
I discuss Landau Hamiltonians with a weak coupling random electric potential of breather type. Under appropriate assumptions a Wegner estimate holds.
It implies the Hölder continuity of the integrated density of states.
The main challenge is the problem how to deal with non-linear dependence on the random parameters.

http://www.mat.uc.cl/~graikov/seminar.html
2018-10-18
17:00hrs.
Georgi Raikov. Pontificia Universidad Católica de Chile
Perturbaciones geométricas de hamiltonianos cuánticos magnéticos
Sala 2
Abstract:

Se considerarán algunas perturbaciones geométricas del operador de Schrödinger tridimensional con campo magnético constante. Se introducirá la función de corrimiento espectral (spectral shift function) y se discutirá su comportamiento asintótico cerca de los niveles de Landau que tienen rol de umbrales para el operador  no perturbado.

http://www.mat.uc.cl/~graikov/seminar.html
2018-08-09
17:00hrs.
Frédéric Klopp. Institut de Mathématiques Jussieu - Paris Rive Gauche, Sorbonne
Resonances for large random systems
Sala 1
Abstract:
The talk is devoted to the description of the resonances generated by a large sample of random material. In one dimension, one obtains a very precise description for the resonances that are directly related to the description for the eigenvalues and localization centers for the full random model. In higher dimension, below a region of localization in the spectrum for the full random model, one computes the asymptotic density of resonances in some sub exponentially small strip below the real axis. This talk is partially based on joint work with M. Vogel.
2018-06-21
17:00hrs.
2018-05-24
17:00hrs.
One-channel operators, a general radial transfer matrix approach and absolutely continuous spectrum
sala 1
Abstract:
First I will introduce one-channel operators and their spectral theory analyses through transfer matrices solving the eigenvalue equation.
Then, inspired from the specific form of these transfer matrices, we will define sets of transfer matrices for any discrete Hermitian operator with locally finite hopping by considering quasi-spherical partitions.
A generalization of some spectral averaging formula for Jacob operators is given and criteria for the existence and pureness of absolutely continuous spectrum are derived.
In the one-channel case this already led to several examples of existence of absolutely continuous spectrum for the Anderson models on such graphs with finite dimensional growth (of dimension $d>2$).
The method has some potential of attacking the open extended states conjecture for the Anderson model in $\mathbb{Z}^d, d\geq 3$.
2018-05-17
17:00hrs.
The Spectral Theorem in the Study of the Fractional Schrödinger Equation
Sala 1
Abstract:
We study the linear fractional Schrödinger equation on a Hilbert space, with a fractional time derivative.  Using the spectral theorem we prove existence and uniqueness of strong solutions, and we show that the solutions are governed by an operator solution family. Examples will be discussed.
http://www.mat.uc.cl/~graikov/seminar.html
2018-05-03
17:00hrs.
Georgi Raikov. Pontificia Universidad Católica de Chile
Lifshits tails for randomly twisted quantum waveguides
Sala 1
Abstract:
I will consider the Dirichlet Laplacian on a three-dimensional twisted waveguide with random Anderson-type twisting. I will discuss the Lifshits tails for the related integrated density of states  (IDS), i.e. the asymptotics of the IDS as the energy approaches from above the infimum of its support. In particular, I will specify  the dependence of the Lifshits exponent on the decay rate of the single-site twisting.
The talk is based on joint works with Werner Kirsch (Hagen) and David Krejcirik (Prague).

2018-03-27
17:00hrs.
Rajinder Mavi. Michigan State University
Anderson localization for a disordered polaron
Abstract:
We will consider an operator modeling a tracer particle on the integer lattice subject to an Anderson field, we associate a one dimensional oscillator to each site of the lattice. This forms a polaron model where the oscillators communicate only through the hopping of the tracer particle. This introduces, a priori, infinite degeneracies of bare energies at large distances. We nevertheless show Dynamical Localization of the tracer particle for compact subsets of the spectrum.

This is joint work with Jeff Schenker.
2018-03-22
17:00hrs.
Timo Weidl . Universität Stuttgart
Sharp semiclassical estimates with remainder terms
Sala 1
Abstract:
Sharp semi-classical spectral estimates give uniform bounds on eigenvalue sums in terms of their Weyl asymptotics. Famous examples are the Li-Yau and the Berezin inequalities on eigenvalues of the Dirichlet Laplacian in domains. Recently these bounds have been sharpened with additional remainder terms, as in the Melas inequality. I give an overview on some of these results and, in particular, I will talk on a Melas type bound for the two-dimensional Dirichlet Hamiltonian with constant magnetic field in a bounded domain.
2017-11-30
17:00hrs.
Diomba Sambou. Facultad de Matemáticas, PUC
On the discrete spectrum of non-self-adjoint Pauli operators with non constant magnetic fields
Sala 1
Abstract:
I will talk about the discrete spectrum generated by complex matrix-valued perturbations for a class of 2D and 3D Pauli operators with non-constant admissible magnetic fields. We shall establish a simple criterion for the potentials to produce discrete spectrum near the low ground energy of the operators. Moreover, in case of creation of non-real eigenvalues, this criterion specifies also their location.

http://www.mat.uc.cl/~graikov/seminar.html
2017-11-23
17:00hrs.
Pablo Miranda. Universidad de Santiago de Chile
Resonancias en guías de ondas torcidas
Sala 1
Abstract:
En esta charla consideraremos el Laplaciano definido en  una guía de ondas recta, la cual será  torcida localmente. Se sabe que tal perturbación no crea  valores propios discretos. Sin embargo, es posible definir una extensión meromorfa de la resolvente del Laplaciano perturbado, la que nos permite  mostrar que existe exactamente una resonancia cerca de ínfimo del espectro esencial.  Para esta resonancia calcularemos su   comportamiento asintótico, en función del tamaño del torcimiento. Por último daremos una idea de como extender estos resultados para los  "umbrales" superiores en el espectro del Laplaciano no perturbado.
2017-11-16
17:00hrs.
Siegfried Beckus. Technion, Haifa, Israel
Shnol type Theorem for the Agmon ground state
Sala 1
Abstract:
The celebrated Shnol theorem asserts that every polynomially bounded generalized eigenfunction for a given energy E associated with a Schrodinger operator H implies that E is in the L2-spectrum of H. Later Simon rediscorvered this result independently and proved additionally that the set of energies admiting a polynomially bounded generalized eigenfunction is dense in the spectrum. A remarkable extension of these results hold also in the Dirichlet setting. It was conjectured that the polynomial bound on the generalized eigenfunction can be replaced by an object intrinsically defined by H, namely, the Agmon ground state. During the talk, we positively answer the conjecture indicating that the Agmon ground state describes the spectrum of the operator H. Specifically, we show that if u is a generalized eigenfunction for the eigenvalue E that is bounded by the Agmon ground state then E belongs to the L2-spectrum of H. Furthermore, this assertion extends to the Dirichlet setting whenever a suitable notion of Agmon ground state is available.

http://www.mat.uc.cl/~graikov/seminar.html
2017-11-09
17:00hrs.
Vincent Bruneau. Université de Bordeaux, France
Spectral analysis in the large coupling limit for singular perturbations
Sala 1
Abstract:
We consider a singular perturbation of the Laplacian, supported on a bounded domain with a large coupling constant.
We study the asymptotic behavior of spectral quantities (eigenvalues and resonances) when the coupling constant tends to infinity.
Joint work with G. Carbou.

http://www.mat.uc.cl/~graikov/seminar.html
2017-10-26
17:00hrs.
Dr. Guo Chuan Thiang . University of Adelaide
Time-reversal, monopoles, and equivariant topological matter
Sala 1
Abstract:
A crucial feature of experimentally discovered topological insulators (2008) and semimetals (2015) is time-reversal, which realises an order-two symmetry "Quaternionically''. Guided by physical intuition, I will formulate a certain equivariant Poincare duality which allows a useful visualisation of "Quaternionic'' characteristic classes and the concept of Euler structures. I also identify a new monopole with torsion charge, and show how the experimental signature of surface Fermi arcs are holographic versions of bulk Dirac strings.
http://www.mat.uc.cl/~graikov/seminar.html
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.

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).