# Seminario de Teoría Espectral

2019-10-24
17:00hrs.
Nora Doll. Friedrich-Alexander University of Erlangen
Connecting Real index pairings to spectral flows
Sala 2
Abstract:
Object of this talk are index pairings of a projection and an unitary where both, the projection and the unitary fulfill Real symmetry relations. For a given combination of  symmetries the Noether index of the pairing vanishes, but there may be a Z_2-index given by the dimension of the kernel, modulo 2. Aim of this talk is to give an interpretation of these Z_2-indices as spectral flows. Two applications are given, one concerning topological insulators, the other concerning the Spectral Localizer.
2019-06-27
17:00hrs.
Leonid Parnovski. University College London
Floating mats and sloping beaches: spectral asymptotics of the Steklov problem on polygons.
Sala 2
Abstract:
I will discuss recent results (joint with M.Levitin, I.Polterovich and D.Sher) on the asymptotic behaviour of Steklov eigenvalues on polygons and other two-dimensional domains with corners. The answer is completely unexpected and depends on the arithmetic properties of the angles.

2019-06-06
17:00hrs.
Horia Cornean. Aalborg University, Denmark
Peierls' substitution for low lying spectral energy windows
Sala 2
Abstract:
We consider a 2d periodic Schrödinger operator for which we assume that either the first Bloch eigenvalue remains isolated while its corresponding Riesz spectral projection family has a non-zero Chern number, or the first two Bloch eigenvalues have a conical crossing. The system is afterwards  perturbed by a weak magnetic field which slowly varies around a positive mean. Then we prove the appearance of a “Landau type” structure of spectral islands and gaps both at the bottom of the spectrum, and near the possible crossings.

This is joint (past and ongoing) work with B. Helffer (Nantes) and R. Purice (Bucharest).
http://www.mat.uc.cl/~graikov/seminar.html
2019-05-30
17:00hrs.
Dmitrii Shirokov. National Research University Higher School of Economics, Russia
On constant solutions of SU(2) Yang-Mills equations
Sala 2
Abstract:
We present all constant solutions of the Yang-Mills equations with SU(2) gauge symmetry for an arbitrary constant non-Abelian current in Euclidean space of arbitrary finite dimension. We use the singular value decomposition method and the method of two-sheeted covering of orthogonal group by spin group to do this. Using hyperbolic singular value decomposition, we solve the same problem in arbitrary pseudo-Euclidean space. The case of Minkowski space is discussed in details. Nonconstant solutions of the Yang-Mills equations are considered in the form of series of perturbation theory.

http://www.mat.uc.cl/~graikov/seminar.html
2019-05-16
17:00hrs.
Frédéric Klopp. Institut de Mathématiques Jussieu - Paris Rive Gauche, Sorbonne
Exponential decay for the 2 particle density matrix of disordered many-body fermions at zero and positive temperature.
Sala 2
Abstract:
We will consider a simple model for interacting fermions in a random background at zero and positive temperature. At zero temperature, we prove exponential decay for the 2 particle density matrix of a ground state. At positive temperature we prove exponential decay for the 2 particle density matrix of the density operator in the grand canonical ensemble.
http://www.mat.uc.cl/~graikov/seminar.html
2019-05-02
17:00hrs.
Georgi Raikov. Pontificia Universidad Católica de Chile
Threshold singularities of the spectral shift function for geometric perturbations of a magnetic Hamiltonian II
Sala 2
Abstract:
I will consider the 3D Schrödinger operator $H_0$ with constant magnetic field, and its perturbations $H_+$ (resp., $H_-$ ) obtained from $H_0$ by imposing Dirichlet (resp., Neumann) conditions on an appropriate surface. I will introduce the Krein spectral shift function for the operator pairs $( H_+, H_0)$ and $(H_-, H_0)$, and will discuss its singularities at the Landau levels which play the role of thresholds in the spectrum of the unperturbed operator $H_0$.

The talk is based on a joint work with V. Bruneau (Bordeaux).
http://www.mat.uc.cl/~graikov/seminar.html
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