**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 H_{0}
with constant magnetic field, and its perturbations H_{+}
(resp., H_{−} )
obtained from H_{0} by
imposing Dirichlet (resp., Neumann) conditions on an appropriate
surface. I will introduce the Krein spectral shift function for the
operator pairs (H_{+,}H_{0})
and (H_{−,}H_{0}),
and will discuss its singularities at the Landau levels which play
the role of thresholds in the spectrum of the unperturbed operator
H_{0}.

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**,

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},
*d*
≥ 3.

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 stateSiegfried
Beckus, Technion, Haifa, Israel**

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 operatorsJake
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**