Spectral Theory Seminar

Pontificia Universidad Católica de Chile, Campus San Joaquín 
Vicuña Mackenna 4860, Facultad de Matemáticas, Sala 1
Thursday, 17:00 - 18:30

 

30 November 2017: On the discrete spectrum of non-self-adjoint Pauli operators with non constant magnetic fields

Diomba Sambou, Facultad de Matemáticas, PUC

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.

 

 

23 November 2017: Resonancias en Guías de Ondas Torcidas

Pablo Miranda, Universidad de Santiago de Chile

Resumen:
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 del ínfimo del espectro esencial. Para esta resonancia calcularemos su comportamiento asintótico, en función del tamaño del torcimiento. Por último daremos una idea de cómo extender estos resultados para los "umbrales" superiores en el espectro del Laplaciano no perturbado.

 

 

16 November 2017: Shnol type theorem for the Agmon ground state
Siegfried Beckus, Technion, Haifa, Israel
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.

 

9 November 2017: Spectral analysis in the large coupling limit for singular perturbations

Vincent Bruneau, Université de Bordeaux, France

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.

 

26 October 2017: Time-reversal, monopoles, and equivariant topological matter

Guo Chuan Thiang, University of Adelaide, Australia

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.

 

28 September 2017: Teoría espectral de hamiltonianos cuánticos. II

Georgi Raikov, Facultad de Matemáticas, PUC

Resumen

En la primera charla se presentaron algunos hechos básicos como el teorema espectral y se dió una idea de cuantización.

En la segunda charla se continuará con la cuantización, se discutirán las propiedades básicas del operador de Schrödinger y también temas más avanzados como la teoría de operadores pseudodiferenciales que juegan el rol de hamiltonianos cuánticos.

Bibliografía

 

21 September 2017: Teoría espectral de hamiltonianos cuánticos. I

Georgi Raikov, Facultad de Matemáticas, PUC

Resumen

El propósito principal del ciclo de dos charlas "Teoría espectral de hamiltonianos cuánticos" es despertar el interés de alumnos de licenciatura hacia los problemas matemáticos que aparecen en la teoría espectral de hamiltonianos cuánticos (operadores auto-adjuntos en espacios de Hilbert).

En la primera charla se presentarán algunos hechos básicos como el teorema espectral, la idea de cuantificación y las propiedades básicas de los operadores de Schrödinger, Pauli y Dirac.

En la segunda charla se hablará de temas más avanzados como hamiltonianos fibrados (por ejemplo, operadores con coeficientes periódicos) y hamiltonianos ergódicos.

Como ya indicado, el ciclo está orientado hacia alumnos de licenciatura pero puede ser interesante también para alumnos de postgrado y postdoctorados.

 

10 August 2017: A Hardy-Lieb-Thirring inequality for fractional Pauli operators

Soeren Fournais, Aarhus Universiy

Abstract

 

18 May 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 April 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 March 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 March 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 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 January 2017: Ballistic Propagation for limit-periodic Jacobi operators
Jake Fillman, Virginia Tech, USA

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