Spectral Theory and PDE Seminar

Pontificia Universidad Católica de Chile, Campus San Joaquín
Vicuña Mackenna 4860, Facultad de Matemáticas, Sala 2
Thursday, 17:00 - 18:30

27 June 2019: Floating mats and sloping beaches: spectral asymptotics of the Steklov problem on polygons

Leonid Parnovski, University College London

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.

6 June 2019: Peierls' substitution for low lying spectral energy windows

Horia Cornean, Aalborg University

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).

30 May 2019: On Constant Solutions of Su(2) Yang-Mills Equations

Dmitrii Shirokov, National Research University Higher School of Economics, Russia

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 decomposition 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.

16 May 2019: Exponential decay for the 2 particle density matrix of disordered many-body fermions at zero and positive temperature

Frédéric Klopp, Institut de Mathématiques Jussieu - Paris Rive Gauche, Sorbonne

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.

02 May 2019: Threshold singularities of the spectral shift function for geometric perturbations of a magnetic Hamiltonian II

Georgi Raikov, Facultad de Matemáticas, PUC

Abstract:

I will consider the 3D Schrödinger operator H0 with constant magnetic field, and its perturbations H+ (resp., H ) obtained from H0 by imposing Dirichlet (resp., Neumann) conditions on an appropriate surface. I will introduce the Krein spectral shift function for the operator pairs (H+,H0) and (H−,H0), and will discuss its singularities at the Landau levels which play the role of thresholds in the spectrum of the unperturbed operator H0.
The talk is based on a joint work with V. Bruneau (Bordeaux).

13 December 2018: Wegner estimate for Landau-breather Hamiltonians

Ivan Veselic, TU Dortmund

Abstract:
I discuss Landau Hamiltonians with a weak coupling random electric potential of breather type. Under appropriate assumptions a Wegner estimate holds.
It implies the Hölder continuity of the integrated density of states.
The main challenge is the problem how to deal with non-linear dependence on the random parameters.

18 October 2018: Perturbaciones Geométricas de Hamiltonianos Cuánticos Magnéticos

Georgi Raikov, Facultad de Matemáticas, PUC

Resumen:

Se considerarán algunas perturbaciones geométricas del operador de Schrödinger tridimensional con campo magnético constante. Se introducirá la función de corrimiento espectral (spectral shift function) y se discutirá su comportamiento asintótico cerca de los niveles de Landau que tienen rol de umbrales para el operador no perturbado.

20-22 September 2018: Chile-Japan Workshop on Mathematical Physics and Partial Differential Equations,

# Graduate School of Mathematical Sciences, University of Tokyo.

9 August 2018: Resonances for Large Random Systems

Frédéric Klopp, Institut de Mathématiques Jussieu - Paris Rive Gauche, Sorbonne

Abstract:
The talk is devoted to the description of the resonances generated by a large sample of random material. In one dimension, one obtains a very precise description for the resonances that are directly related to the description for the eigenvalues and localization centers for the full random model. In higher dimension, below a region of localization in the spectrum for the full random model, one computes the asymptotic density of resonances in some sub exponentially small strip below the real axis. This talk is partially based on joint work with M. Vogel.

21 June 2018: Semiclassical Trace Formula and Spectral Shift Function for Schrödinger Operators with Matrix-Valued Potentials

Marouane Assal, Facultad de Matemáticas, PUC

Abstract:

In this talk, I will present some recent results on the spectral properties of semiclassical systems of pseudodifferential operators. We developed a stationary approach for the study of the Spectral Shift Function for a pair of self-adjoint semiclassical Schrödinger operators with matrix-valued potentials. A Weyl-type semiclassical asymptotics with sharp remainder estimate for the SSF is obtained, and under the existence condition of a scalar escape function, a full asymptotic expansion for its derivatives is proved. This last result is a generalization of the result of Robert-Tamura (1984) proved in the scalar case near non-trapping energies. Our results are consequences of semiclassical trace formulas for general microhyperbolic systems possibly with eigenvalues crossings.

This talk is based on a recent work with Mouez Dimassi (University of Bordeaux, France) and Setsuro Fujiié (Ritsumeikan University, Japan).

24 May 2018: One-Channel Operators, a General Radial Transfer Matrix Approach and Absolutely Continuous Spectrum

Abstract:

First I will introduce one-channel operators and their spectral theory analyses through transfer matrices solving the eigenvalue equation. Then, inspired from the specific form of these transfer matrices, we will define sets of transfer matrices for any discrete Hermitian operator with locally finite hopping by considering quasi-spherical partitions. A generalization of some spectral averaging formula for Jacob operators is given and criteria for the existence and pureness of absolutely continuous spectrum are derived. In the one-channel case this already led to several examples of existence of absolutely continuous spectrum for the Anderson models on such graphs with finite dimensional growth (of dimension d > 2). The method has some potential of attacking the open extended states conjecture for the Anderson model in Zd, d3.

17 May 2018: The Spectral Theorem in the Study of the Fractional Schrödinger Equation

Abstract:

We study the linear fractional Schrödinger equation on a Hilbert space, with a fractional time derivative. Using the spectral theorem we prove existence and uniqueness of strong solutions, and we show that the solutions are governed by an operator solution family. Examples will be discussed.

3 May 2018: Lifshits tails for randomly twisted quantum waveguides

Georgi Raikov, Facultad de Matemáticas, UC

Abstract:

I will consider the Dirichlet Laplacian on a three-dimensional twisted waveguide with random Anderson-type twisting. I will discuss the Lifshits tails for the related integrated density of states (IDS), i.e. the asymptotics of the IDS as the energy approaches from above the infimum of its support. In particular, I will specify the dependence of the Lifshits exponent on the decay rate of the single-site twisting.
The talk is based on joint works with Werner Kirsch (Hagen) and David Krejcirik (Prague).

27 March 2018: Anderson localization for a disordered polaron

Rajinder Mavi, Michigan State University

Abstract:

We will consider an operator modeling a tracer particle on the integer lattice subject to an Anderson field, we associate a one dimensional oscillator to each site of the lattice. This forms a polaron model where the oscillators communicate only through the hopping of the tracer particle. This introduces, a priori, infinite degeneracies of bare energies at large distances. We nevertheless show Dynamical Localization of the tracer particle for compact subsets of the spectrum.

This is joint work with Jeff Schenker.

22 March 2018: Sharp semiclassical estimates with remainder terms

Timo Weidl, Universität Stuttgart

Abstract:

Sharp semi-classical spectral estimates give uniform bounds on eigenvalue sums in terms of their Weyl asymptotics. Famous examples are the Li-Yau and the Berezin inequalities on eigenvalues of the Dirichlet Laplacian in domains. Recently these bounds have been sharpened with additional remainder terms, as in the Melas inequality. I give an overview on some of these results and, in particular, I will talk on a Melas type bound for the two-dimensional Dirichlet Hamiltonian with constant magnetic field in a bounded domain.

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.

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.