Category: Seminar Spectral Theory and PDE

Speaker: Leonid Parnovski. University College London

Title: Floating mats and sloping beaches: spectral asymptotics of the Steklov problem on polygons.

Place: PUC Chile, Campus San Joaquin, Fac. Matematicas, 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. 

Expositor:  Horia Cornean, Aalborg University

Title:  Peierls’ substitution for low lying spectral energy windows 

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

Speaker: Dmitrii Shirokov. National Research University Higher School of Economics, Russia

Title: On Constant Solutions of Su(2) Yang-Mills Equations

Place: Pontificia Universidad Católica, Facultad de Matemáticas (Campus San Joaquin), 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 decomposion 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 

Frédéric Klopp. Institut de Mathématiques Jussieu – Paris Rive Gauche, Sorbonne
Title: Exponential decay for the 2 particle density matrix of disordered many-body fermions at zero and positive temperature.
Place: Pontificia Universidad Católica de Chile, Facultad de Matemáticas (campus San Joaquin), 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

Georgi Raikov. Pontificia Universidad Católica de Chile
Title: Threshold singularities of the spectral shift function for geometric perturbations of a magnetic Hamiltonian
Place: Pontificia Universidad Católica de Chile, Facultad de Matemáticas (campus San Joaquin), Sala 2
Abstract:
I will consider the 3D Schrödinger operator H0 with constant magnetic field, and its perturbations  (resp.,  ) obtained from   by imposing Dirichlet (resp., Neumann) conditions on an appropriate surface. I will introduce the Krein spectral shift function for the operator pairs  and , and will discuss its singularities at the Landau levels which play the role of thresholds in the spectrum of the unperturbed operator . 
The talk is based on a joint work with V. Bruneau (Bordeaux).
http://www.mat.uc.cl/~graikov/seminar.html