**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`

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**,

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**

**Christian Sadel,
Facultad de Matemáticas, UC**

**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 **Z*** ^{d}*,

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

**Humberto
Prado, Universidad de Santiago de Chile**

**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
**

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.

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**

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 ** Z_{2}**-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
**

**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**** **