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.
Radu
Purice, Instutute of
Mathematics, Romanian Academy of Sciences
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 Wigners 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:
Revisamos resultados anteriores del
operador de Pauli 2D: el Teorema de Aharonov-Casher y resultados acerca del
operador de campos magnéticos periódicos.
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
Christian Sadel, Facultad
de Matemáticas, PUC
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 states 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
Marius Mantoiu, Facultad
de Ciencias, Universidad de Chile
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.
Seminarios 2013-2014; Seminarios 2010-2012;
Seminarios 2008-2009