Seminario de Teoría Espectral

Pontificia Universidad Católica de Chile, Campus San Joaquín

Vicuña Mackenna 4860, Facultad de Matemáticas, Sala 1

Jueves, 17:00 - 18:30

5 de enero de 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.

27 de diciembre de 2016, 15:00: Modeling thermostats using master equations

Michael Loss, Georgia Institute of Technology, Atlanta, USA

Abstract:

In this talk we discuss results for a model of randomly colliding particles interacting with a thermal bath, i.e., a thermostat. Collisions between particles are modeled via the Kac master equation while the thermostat is seen as an infinite gas at thermal equilibrium with inverse temperature β. The evolution propagates chaos and the one particle marginal, in the limit of large systems, satisfies an effective Boltzmann-type equation. The system admits the canonical distribution at inverse temperature β as the unique equilibrium state. It turns out that any initial distribution approaches the equilibrium distribution exponentially fast, both, in a proper function space as well as in relative entropy. Recent results concerning the approximation of thermostats by a large but finite heat reservoir will also be discussed. It turns out that in suitable norms the approximation can be shown to be uniformly in time, i.e., the error depends only on the size of the finite heat reservoir. This is joint work with Federico Bonetto, Hagop Tossounian and Ranjini Vaidyanathan.

22 de diciembre de 2016, 14:00: Eigensystem multiscale analysis for Anderson localization in energy intervals

Abel Klein, University of California, Irvine, USA

Abstract:

We present an eigensystem multiscale analysis for proving localization (pure point spectrum with exponentially decaying eigenfunctions, dynamical localization) for the Anderson model in an energy interval.  In particular, it yields localization for the Anderson model in a nonempty interval at the bottom of the spectrum. This eigensystem multiscale analysis in an energy interval treats all energies of the finite volume operator at the same time, establishing level spacing and localization of eigenfunctions with eigenvalues in the energy interval in a fixed box with high probability.  In contrast to the usual strategy, we do not study finite volume Green's functions. Instead, we perform a multiscale analysis based on finite volume eigensystems (eigenvalues and eigenfunctions). In any given scale we only have decay for eigenfunctions with eigenvalues in the energy interval, and no information about the other eigenfunctions.  For this reason, going to a larger scale requires new arguments that were not necessary in our previous eigensystem multiscale analysis for the Anderson model at high disorder, where in a given scale we have decay for all eigenfunctions. (Joint work with A. Elgart)

1 de diciembre de 2016, 15:30: Spectral gaps for periodic Hamiltonians in slowly varying magnetic fields.

Abstract:
I report on some work done in collaboration with H. Cornean and B. Helffer. We consider a periodic Schroedinger operator in two dimensions, perturbed by a weak magnetic field whose intensity slowly varies around a positive mean. We show in great generality that the bottom of the spectrum of the corresponding magnetic Schroedinger operator develops spectral islands separated by gaps, reminding of a Landau-level structure.

First, we construct an effective magnetic matrix which accurately describes the low lying spectrum of the full operator. The construction of this effective magnetic matrix does not require a gap in the spectrum of the non-magnetic operator, only that the first and the second Bloch eigenvalues never cross.

Second, we perform a detailed spectral analysis of the effective matrix using a gauge-covariant magnetic pseudo-differential calculus adapted for slowly varying magnetic fields.

1 de diciembre de 2016, 17:00: Ground state energy of the Robin Laplacian in corner domains.

Nicolas Popoff, University of Bordeaux, France

Abstract:
I will consider the problem of the asymptotics of the first eigenvalue for the Laplacian with Robin boundary condition, when the Dirichlet parameter gets large. I will focus on the case where the domain belongs to a general class of corner domains, and show that the asymptotics is given at first order by the minimization of a function, called "local energy", defined on the tangent geometries. A key quantity of our analysis is the infimum of the essential spectrum of the Robin Laplacian on a cone.  Then, using a multiscale analysis, we give an estimate of the remainder. I will also provide a more precise asymptotics when the domain is regular, using a semiclassical effective Hamiltonian defined on the boundary and involving the mean curvature.

24 de noviembre de 2016: On the eigenvalue distribution measure for random matrices and random Schroedinger operators

Werner Kirsch, Hagen University, Germany

Abstract:

We discuss classical and recent results on the distribution of eigenvalues (density of states) for random matrices and compare them to results for random Schroedinger operators.

We discuss Wigner’s semicircle law and some of its generalizations and sketch a rather elementary proof.

17 de noviembre de 2016: Valores propios nulos del operador de Pauli 2D de campos magnéticos casi periódicos.

Nicolás Espinoza, Facultad de Matemáticas, PUC

Abstract:

Luego revisamos el problema para un campo magnético casi periódico y describimos el Kernel de un campo magnético particular.

10 de noviembre de 2016: The principal trace formula and its applications to index theory, spectral shift function and spectral flow

Galina Levitina, UNSW, Australia

Abstract:
Let {A(t)}_{t\in R} be a one parameter family of self-adjoint operators on a separable Hilbert space H, that converges in norm resolvent sense to the asymptotes A_\pm. Consider the operator D_A=d/dt+A(t) on the Hilbert space L_2(R,H). Without any assumption on the spectra of the operators A_\pm we prove trace formula for semigroup difference of D_A, which was proved initially by Robbin-Salamon under the assumption of purely discrete spectra of A_\pm. As a consequence of this trace formula we establish the connection between spectral shift function for the pair of the asymptotes (A_+,A_-), index theory for the operator D_A and the spectral flow for the family {A(t)}_{t\in R} applicable for differential operators in higher dimensions. This talk is the significant extension of the results presented by Prof. Fedor Sukochev on the conference 'Spectral theory and Mathematical Physics', Santiago, 2014.

3 de noviembre de 2016: A novel way of constructing Hadamard state in absence of symmetry

Simone Murro, University of Regensburg, Germany

Abstract:

​​We give a functional analytic construction of algebraic states for CAR algebras on a globally hyperbolic Lorentzian manifold. We show that in Minkowski space we recover the vacuum state and when we couple the Dirac equation to a time-dependent external potential, which is smooth and decays faster than quadratically for large times, we obtain Hadamard states.

29 de septiembre de 2016: Comportamiento asintótico de los valores propios pequeños del Laplaciano de Krein perturbado

Georgi Raikov, Facultad de Matemáticas, PUC

Resumen:

Consideraremos el Laplaciano de Krein K en un dominio acotado regular, perturbado por un multiplicador real V, que se anula en la frontera.

Suponiendo que V tiene un signo definido, vamos a discutir el comportamiento asintótico de la sucesión de valores propios de K+V que tiende al origen. En particular, vamos a demostrar que el Hamiltoniano efectivo que determina el término asintótico principal, es el operador armónico de Toeplitz con símbolo V, unitariamente equivalente a un operador pseudodiferencial en la frontera.

Se trata de un trabajo en conjunto con Vincent Bruneau (Burdeos, Francia).

15 de septiembre de 2016: Singularidades de la Función de Corrimiento Espectral para un Hamiltoniano Magnético en el Semiplano.

Pablo Miranda, Facultad de Matemáticas, PUC

Resumen:

En esta charla consideraremos el operador de Schroedinger H, con  campo magnético constante, definido en un semi-plano y  perturbado por un potencial V que decae al infinito.  Como una posible extensión del problema de conteo de valores propios discretos de H+V, introduciremos  la Función de Corrimiento Espectral. Probaremos que esta función es acotada en conjuntos compactos que no contienen a los valores de Landau y describiremos su comportamiento asintótico en las singularidades que se presentan  en estos valores. Para los resultados mostrados se considerará la condición de borde de Dirichlet. Resultados con la condición de Neumann también será discutidos.

23 de junio de 2016: Degree, mixing, and absolutely continuous spectrum of cocycles with values in compact Lie groups II

Rafael Tiedra de Aldecoa, Facultad de Matemáticas, PUC

Abstract:

This is the second and last talk on degree, mixing, and absolutely continuous spectrum of cocycles with values in compact Lie groups.

We consider skew products transformations T_phi associated to cocycles phi with values in compact Lie groups G.

We define the degree of phi as a suitable function on the base space, we show that the degree transforms in a natural way under Lie group homomorphisms and under the relation of C^1-cohomology, and we explain how it generalises previous definitions of degree of a cocycle. For each finite-dimensional irreducible representation pi of G, we define in an analogous way the degree of pi°phi. Under ergodicity assumptions, we show that the degree of phi reduces to a constant given by an integral (average). As a by-product, we obtain that there is no uniquely ergodic skew product T_phi with nonzero degree if G is a connected semisimple compact Lie group.

Next, we show that T_phi is mixing in the orthocomplement of the kernel of of the degree of pi°phi, and under some additional assumptions we show that T_phi has purely absolutely continuous spectrum in that orthocomplement. Summing up these results for each pi, one obtains a global result for the mixing and the absolutely continuous spectrum of T_phi. As an application, we present two explicit cases: when G is a torus and when G=U(2).

Our proofs rely on new results on positive commutator methods for unitary operators in Hilbert spaces.

16 de junio de 2016: Degree, mixing, and absolutely continuous spectrum of cocycles with values in compact Lie groups

Rafael Tiedra de Aldecoa, Facultad de Matemáticas, PUC

Abstract:

We consider skew products transformations T_phi associated to cocycles phi with values in compact Lie groups G.

We define the degree of phi as a suitable function on the base space, we show that the degree transforms in a natural way under Lie group homomorphisms and under the relation of C^1-cohomology, and we explain how it generalises previous definitions of degree of a cocycle. For each finite-dimensional irreducible representation pi of G, we define in an analogous way the degree of pi°phi. Under ergodicity assumptions, we show that the degree of phi reduces to a constant given by an integral (average). As a by-product, we obtain that there is no uniquely ergodic skew product T_phi with nonzero degree if G is a connected semisimple compact Lie group.

Next, we show that T_phi is mixing in the orthocomplement of the kernel of of the degree of pi°phi, and under some additional assumptions we show that T_phi has purely absolutely continuous spectrum in that orthocomplement. Summing up these results for each pi, one obtains a global result for the mixing and the absolutely continuous spectrum of T_phi. As an application, we present two explicit cases: when G is a torus and when G=U(2).

Our proofs rely on new results on positive commutator methods for unitary operators in Hilbert spaces.

(This is a two-session talk.)

5 de mayo de 2016: SDE limits for products of random matrices and GOE statistics for rescaled Anderson models on long strips

Abstract:

We consider products of random i.i.d. perturbations T_n of a fixed matrix T and consider the products T_n ... T_1 where the random perturbations have a specific scaling with n. Projecting out exponential growing terms by a Schur complement and normalising fast rotations, we obtain a limiting process for these products which is described by an SDE. Applying this to transfer matrices for random Schrödinger operators on strips in Z^2 we get a random matrix limit for the eigenvalue process along certain sequences of strips.

28 de abril de 2016: Sobre teoremas de distribución límite de autovalores para el hidrógeno en campos eléctricos o magnéticos constantes

Carlos Villegas-Blas, Universidad Nacional Autónoma de México

Resumen:

Consideremos el Hamiltoniano del átomo de hidrógeno inmerso ya sea en un campo eléctrico o magnético constante.  Debido a teoremas de estabilidad de Avron-Herbst-Simon,  es posible definir cúmulos de resonancias o autovalores considerando campos suficientemente débiles respectivamente. En esta charla describiremos el estudio de la manera en que ya sea las resonancias o autovalores se distribuyen en los cúmulos en el límite semiclásico (considerando a la constante de Planck adquiriendo  valores discretos adecuados que tienden a cero).

19 de noviembre de 2015: Multidimensional frequency estimation using general domain Hankel and Toeplitz operators

Marcus Carlsson, Lund University, Sweden

Abstract:

General domain Hankel and Toeplitz operators is a class of operators that significantly extends the classical counterparts to several variables. I will discuss various results concerning their structure, positive semidefinitness and finite rank. It turns out that their symbols are then sums of exponential functions, and these operators therefore have a potential for playing a key role in multidimensional frequency estimation / approximation by sparse exponential sums. I will elaborate on this connection and show some numerical results. Potential applications range from seismic imaging to chemistry (NMR) and medicine (e.g. MRI).

12 de noviembre de 2015: On the Mathematics of Political Power

Popular lecture

Werner Kirsch, Hagen University, Germany

Abstract:

In most parliaments and democratic committees each member has just one vote which gives all the members the same voting power. However, there are institutions in which the members do not have the same voting power. Examples are the UN Security Council, the International Monetary Fund and many other supranational organizations.

A similar effect occurs in most bicameral parliamentary systems. In such political systems there is a chamber (usually called Senate or Council) in which the member states have a number of votes or seats somehow depending on the member state’s population. This is the case for instance in the Chilean Senate, the German Bundesrat and the Council of Ministers of the European Union.

In this lecture we try to express voting power in such bodies in mathematical terms. We also develop criteria for a fair representation in bodies which represent states or regions, like the Council of the EU and the Senado in Chile.

5 de noviembre de 2015: Spectral properties of horocycle flows for surfaces of negative curvature

Rafael Tiedra de Aldecoa, Facultad de Matemáticas, PUC

Abstract:

We consider flows, called W^u flows, whose orbits are the unstable manifolds of a codimension one Anosov flow. Under some regularity asumptions, we show that W^u flows have purely absolutely continuous spectrum in the orthocomplement of the constant functions. As a particular case, we obtain that a class of horocycle flows for compact surfaces of (possibly variable) negative curvature have purely absolutely continuous spectrum in the orthocomplement of the constant functions. This generalises recent results on time changes of the classical horocycle flows for compact surfaces of constant negative curvature.

29 de octubre de 2015: Difference equation for the Heckman-Opdam hypergeometric function and its confluent Whittaker limit

Erdal Emsiz, Facultad de Matemáticas, PUC

Abstract:

We will discuss explicit difference equations for the Heckman-Opdam hypergeometric function associated with root systems (a generalization of the Gauss hypergeometric function to various variables). Our method exploits the fact that for discrete spectral values on a (translated) cone of dominant weights the Heckman-Opdam hypergeometric function truncates in terms of Heckman-Opdam Jacobi polynomials. This permits us to derive/prove the desired difference equations in two steps: first for the discrete spectral values by performing a so-called $q\to1$ degeneration of a recently found Pieri formula for the celebrated Macdonald polynomials, and then for arbitrary spectral values upon invoking an analytic continuation argument borrowed from Rösler (based on known growth estimates for the Heckman-Opdam hypergeometric function that enable one to apply Carlson's theorem).
If time permits we will also mention analogous difference equation for the class-one Whittaker function diagonalizing the open quantum Toda chain associated with reduced root systems.

Joint work with Jan Felipe van Diejen (Universidad de Talca).

22 de octubre de 2015: The semi-classical limit of large fermionic systems
Søren Fournais, Aarhus University, Denmark
Abstract:

We study a system of $N$ fermions in the regime where the intensity of the interaction scales as $1/N$ and with an effective semi-classical parameter $\hbar=N^{-1/d}$ where $d$ is the space dimension. For a large class of interaction potentials and of external electromagnetic fields, we prove the convergence to the Thomas-Fermi minimizers in the limit $N\to\infty$. The limit is expressed using many-particle coherent states and Wigner functions. The method of proof is based on a fermionic de Finetti-Hewitt-Savage theorem in phase space and on a careful analysis of the possible lack of compactness at infinity.

This is joint work with Mathieu Lewin and Jan Philip Solovej.

15 de octubre de 2015: Energia del estado fundamental de un sistema de polarones
Rafael Benguria, Instituto de Fisica, PUC

Resumen:

El último problema abierto sobre el sistema de muchos polarones, en la aproximación de Pekar-Tomasevich, es el caso de bosones con la repulsión de Coulomb electrón-electrón de constante de acoplamiento exactamente  "1"  (i.e., el `caso neutral´).
En esta charla presentaré la demostración obtenida recientemente en conjunto con Rupert Frank y Elliott Lieb del hecho que la energía del estado fundamental, para N grande se comporta exactamente como -N^{7/5}, y mostraré cotas inferiores y superiores sobre el coeficiente asintótico, las que coinciden dentro de un factor 2^{2/5}.

24 de septiembre de 2015: Desigualdad Isoperimetrica para el autovalor fundamental de una placa empotrada sometida a compresión

Rafael Benguria, Instituto de Fisica, PUC

Resumen:

En 1995, Nadirashvili demostró la conjetura de Rayleigh para el autovalor más bajo de una placa empotrada, i.e., de todas las placas vibrantes de igual material y de la misma área (con condiciones de borde empotrada), la que tiene la frecuencia fundamental más baja es la placa circular. Ashbaugh y Benguria demostraron en forma independiente la conjetura de Rayleigh y su extensión al problema similar en tres dimensiones. La extensión a dimensión N (N >3) es un problema abierto. Uno puede considerar el problema análogo para una placa empotrada sujeta a compresión. En esta charla daré la demostración del equivalente de la conjetura de Rayleigh para este problema. Este es un trabajo conjunto con Mark S. Ashbaugh (U. Missouri) y R. Mahadevan (U. Concepción).

27 de agosto de 2015: Spectral and scattering properties at thresholds for the Laplacian in a half-space with a periodic boundary condition
Rafael Tiedra de Aldecoa, Facultad de Matemáticas, Pontificia Universidad Católica de Chile

Abstract:
For the scattering system given by the Laplacian in a half-space with a periodic boundary condition, we derive resolvent expansions at embedded thresholds, we prove the continuity of the scattering matrix, and we establish new formulas for the wave operators.

20 de agosto de 2015: The Brezis-Nirenberg Problem on SN, in spaces of fractional dimension

Rafael Benguria, Instituto de Física, PUC

25 de junio de 2015: On the essential spectrum

Abstract:

I shall present decompositions of essential spectra of various type of pseudodifferential operators with anisotropic coefficients.

30 de abril de 2015: Anderson transition at 2 dimensional growth rate on antitrees with normalized edge weights
Christian Sadel, Institute of Sience and Technology, Klosterneuburg, Austria

Abstract:

An antitree is a graph $\mathbb{G}$ consisting of shells $S_n, n\geq 0$ which contain finitely many vertices, such that each vertex in $S_n$ is connected with each vertex in $S_{n+1}$ by an edge. There are no further edges. We are particularly interested in the case where the number of vertices in $S_n$ grows polynomially like $n^{d-1}$ which corresponds to a $d$-dimensional growth rate. The growth is uniform, if $\lim \#(S_n) / n^{d-1}=c>0$ exists.
We normalize the edges to get a bounded adjacency operator $A$ and consider the Anderson model $A+V$ where $V$ is a random, i.i.d. potential. In a certain set $I$ of energies and for uniformly $d$-dimensional growth rates, we obtain a transition in the type of the spectrum from pure point ($d<2$) to partly pure point and partly singular continuous $d=2$) to absolutely continuous $d>2$).

16 de abril de 2015: Discrete spectrum of Schroedinger operators with oscillating decaying potentials

Georgi Raikov, Facultad de Matemáticas, PUC

Abstract:

We consider the Schroedinger operator HηW = -Δ+ ηW, self-adjoint in L2(Rd), d ≥1. Here η is a non constant almost periodic function, while W decays slowly and regularly at infinity. We study the asymptotic behaviour of the discrete spectrum of HηW near the origin, and due to the irregular decay of ηW, we encounter some non semiclassical phenomena; in particular, HηW has less eigenvalues than suggested by the semiclassical intuition.

19 de marzo de 2015: A criterion for the existence of zero modes for the Pauli operator with fastly decaying fields

Hanne van den Bosch, Instituto de Física, PUC

Abstract:
We consider the Pauli operator in R^3 with magnetic fields decaying "sufficiently fast". It is known magnetic fields giving rise to zero modes are rather "rare" in the 3-dimensional case. We give a criterion for the existence of zero modes for a given field. This also implies a lower bound for the Pauli operator with magnetic fields (in the class considered) without zero modes, which in turn allows to deduce Sobolev, Hardy and CLR inequalities for these operators.