`Pontificia Universidad Católica de Chile, Campus San Joaquín `

`Vicuña Mackenna 4860, Facultad de Matemáticas, Sala 1`

Thursday, 16:30 - 18:00

10 de agosto de 2017: **A
Hardy-Lieb-Thirring
inequality for fractional Pauli operators**

**Soeren Fournais, Aarhus Universiy**

18 de mayo de 2017:
**Anderson localization for one-frequency
quasi-periodic block Jacobi operators**

**Silvius Klein, PUC Rio de Janeiro**

**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.

6 de abril de 2017: **The
Time-Dependent Hartree-Fock-Bogoliubov Equations for
Bosons**

**Sébastien
Breteaux****,
Basque Center for Applied Mathematics**

**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 another 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.

23 de marzo de 2017: **Spectral analysis of quantum walks
with an anisotropic coin**

**Rafael Tiedra de Aldecoa, Facultad de Matemáticas, PUC**

**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).

16 de marzo de 2017: **Finite volume calculation of
topological invariants**

**Hermann
Schulz-Baldes****, University of Erlangen, Germany**

**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.

5 de enero de 2017: **Ballistic
Propagation for limit-periodic Jacobi operators
**

**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.

**Seminarios 2015-2016****; ****Seminarios 2013-2014****; ****Seminarios 2011-2012;
Seminarios 2009-2010;
Seminarios 2008**** **