Seminar Spectral Theory and PDE

## Seminar Spectral Theory and PDE, May 16, 17:00

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

seminar FisMat

## FisMat Seminar, May 15, 15:45

Massimo Moscolari. Aalborg University
Title: Beyond Diophantine Wannier diagrams: gap labelling for Bloch-Landau Hamiltonians
Place: Pontificia Universidad Católica, Facultad de Matemáticas (Campus San Joaquin), Sala 5
Abstract:
In 1978 Wannier discovered a Diophantine relation expressing the integrated density of states of a gapped group of bands of the Hofstadter Hamiltonian as a linear function of the magnetic field flux with integer slope. I will show how to extend this relation to a gap labelling theorem for any 2D Bloch-Landau Hamiltonian operator and to certain non-covariant systems having slowly varying magnetic fields. The integer slope will be interpreted as the Chern character of the projection onto the space of occupied states. The talk is based on a joint work with H. Cornean and D. Monaco.

Seminar Spectral Theory and PDE

## Seminar Spectral Theory and PDE, May 2, 17:00

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 $H_+$ (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 $(H_0;H_+)$ and $(H_0;H_-)$, 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

seminar FisMat

## FisMat Seminar, April 24, 15:45

Svetlana Jitomirskaya. University of California, Irvine
Title: Cantor spectrum of a model of graphene in magnetic field
Place: Pontificia Universidad Católica de Chile, Facultad de Matemáticas (Campus San Joaquin), Sala 5
Abstract:
We consider a quantum graph as a model of graphene in magnetic fi elds and give a complete analysis of the spectrum, for all constant fluxes. In particular, we show that if the reduced magnetic flux  through a honeycomb is irrational, the continuous spectrum is an unbounded Cantor set of Lebesgue measure zero and Hausdorff dimension bounded by 1/2.

Based on joint works with S. Becker, R. Han, and also I. Krasovsky

seminar FisMat

## FisMat Seminar, April 17, 15:45

Walter de Siqueira Pedra. University of São Paulo
Thermodynamical Stability and Dynamics of Lattice Fermions with Mean-Field Interactions
Place: Pontificia Universidad Católica, Facultad Matemáticas (Campus San Joaquin) Sala 5
Abstract:
For lattice fermions we study the thermodynamic limit of the time evolution of observables when the corresponding finite-volume Hamiltonians contain mean-field terms (like, e.g., the BCS model). It is well-known that, in general, this limit does not exist in the sense of the norm of observables, but may exist in the strong operator topology associated to a well-chosen representation of the algebra of observables. We proved that this is always the case for any cyclic representation associated to an invariant minimizer of the free energy density, if the Hamiltonians are invariant under translations. Our proof uses previous results on the structure of states minimizing the free energy density of mean-field models along with Lieb-Robinson bounds for the corresponding families of finite-volume time evolutions. This is a joint work with Jean-Bernard Bru, Sébastien Breteaux and Rafael Miada.