Spectral Theory Seminar

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


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

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

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

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 Z2-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
Jake Fillman, Virginia Tech, USA

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