Spectral and Scattering Theory - MAQM

Liste des Contributions

1. Quantum ergodicity for large quantum graphs

Brian Winn

24/06/2024 14:00

3. The Unruh state for massless fermions on Kerr spacetime and its Hadamard property

Dietrich Häfner

24/06/2024 15:00

We give a rigorous definition of the Unruh state in the setting of massless Dirac fields on slowly rotating Kerr spacetimes. This state is a natural state on a spacetime describing an eternal black hole and also appears as a final state in the context of the collapse of a rotating star. We will show that in the union of exterior and interior region the Unruh state is pure and Hadamard....

4. Transition in the Integrated Density of States of a correlated random Schrödinger operator

Constanza Rojas-Molina

24/06/2024 16:30

In 2017, Sabot, Tarrés and Zeng proved a connection between a reinforced random walks with a non-linear sigma-model coming from statistical mechanics, studied by Disertori, Spencer and Zirnbauer. In both models there is the presence of a random Schrödinger operator. We study the Integrated Density of Stated for this model and show it undergoes a phase transition depending on the dimension and...

5. Hankel operators with band spectra and elliptic functions

Alexander Pushnitski

25/06/2024 09:00

I will discuss spectral properties of bounded self-adjoint Hankel operators H, realised as integral operators on the positive semi-axis, that commute with dilations by a fixed factor. In analogy with the spectral theory of periodic Schroedinger operators, the Hankel operators H of this class admit the Floquet-Bloch decomposition, which represents H as a direct integral of certain compact fiber...

6. Spectral analysis of the magnetic Robin Laplacian

Rayan Fahs

25/06/2024 10:30

In this talk, I will discuss the spectral analysis of the Robin Laplacian on a smooth bounded two-dimensional domain in the presence of a magnetic field with moderate and large intensity.

For the moderate case, in the strong coupling limit, when the Robin parameter tends to infinity, I will explore the spectral gap between successive eigenvalues. In the case of the disc domains, I will...

7. Spectral decomposition on the space of flat surfaces: Laplacians and Siegel—Veech Transforms

Jean Lagacé

25/06/2024 11:30

A classical result in spectral theory is that the space of square integrable functions on the modular surface X = SL(2,Z) \ SL(2,R) can be decomposed as the space of Eisenstein series and its orthogonal complements, the cusp forms. The former space corresponds to the spectral projection on the continuous spectrum of the Laplacian on X, and the cusp forms to the projection on the point...

8. Uniform resolvent estimates for a few non-elliptic operators

Haruya Mizutani

25/06/2024 14:30

We will present recent results on uniform weighted resolvent estimates, or equivalently limiting absorption principles with uniform bounds with respect to the spectral parameter, for two non-elliptic differential operators. One is the massless Klein-Gordon operator on the asymptotically Minkowski spacetime with a sufficiently small metric perturbation and the other is the sub-Laplacian on the...

9. Blow-up on a star-graph

Stefan Le Coz

25/06/2024 15:30

We consider a metric star graph endowed with a nonlinear Schrödinger equation with critical nonlinearity. Depending on the mass of the initial datum, the corresponding solution might be global or blow-up in finite time. At the mass-threshold, we construct a solution with arbitrary energy, which blows up in finite time at the vertex of the star graph. The blow-up profile and blow-up speed are...

11. Duistermaat index and eigenvalue interlacing for perturbations in boundary conditions

Gregory Berkolaiko

26/06/2024 09:00

Eigenvalue interlacing is a tremendously useful tool in linear algebra and spectral analysis. In its simplest form, the interlacing inequality states that a rank-one positive perturbation shifts the eigenvalue up, but not further than the next unperturbed eigenvalue. For different types of perturbations, this idea is known as the "Weyl interlacing" (additive perturbations), "Cauchy...

12. Scattering resonances of large quantum graphs

Maxime Ingremeau

26/06/2024 10:30

Open quantum graphs are singular one-dimensional objects, on which waves can escape towards infinity. The relevant spectral information to describe the behavior of such waves are the scattering resonances of the graph, which form an (infinite) set of complex numbers.
We will state old and new results concerning the location of these resonances, in particular in the asymptotic regime where the...

13. Spectral comparison results on quantum graphs

Joachim Kerner

26/06/2024 11:30

By now, quantum and metric graphs have become popular models in different areas
of mathematics and other areas of science such as physics. Being a (typically) complex
structure which is locally one-dimensional, they in some sense interpolate between one-
and higher-dimensional aspects known, for example, from the study of manifolds. In this
talk, our main goal is to compare the spectrum of...

14. On the discrete eigenvalues of Schrödinger operators with complex potentials

Sabine Boegli

27/06/2024 09:00

In this talk I shall present constructions of Schrödinger operators with complex-valued potentials whose spectra exhibit interesting properties. One example shows that for sufficiently large $p$, the discrete eigenvalues need not be bounded in modulus by the $L^p$ norm of the potential. This is a counterexample to the Laptev-Safronov conjecture (Comm. Math. Phys. 2009). Another construction...

15. Energy decay for strongly damped wave equations

Borbala Gerhat

27/06/2024 10:30

For wave equations with damping unbounded at infinity, essential spectrum may cover the whole negative semi-axis. One can thus not expect the semigroup norm to decay exponentially in time and a more delicate analysis needs to be performed. We derive bounds for the resolvent norm (between suitable spaces) along the imaginary axis and thereby obtain the corresponding polynomial decay rates of...

16. Some applications of the (complexified) metaplectic group to Schrödinger evolutions with non-self-adjoint operators

Joseph Viola

27/06/2024 11:30

We describe how to use the metaplectic group and its complexifications to follow wave packets under the Schrödinger evolution of quadratic operators. In particular, we look at applications to non-self-adjoint operators such as the Davies operator or the circle model for the hypoelliptic Laplacian.

17. A quantum graph approach to metamaterial design

Gregor Tanner

27/06/2024 14:30

Tristan M. Lawrie, Gregor Tanner
School of Mathematical Sciences, University of Nottingham

and

Gregory J Chaplain
School of Physics and Astronomy, University of Exeter

We consider a quantum graph approach for designing metamaterials. An infinite square periodic quantum graph, constructed from vertices and edges, acts as a paradigm for a 2D metamaterial. Wave transport occurs along...

18. High contrast elliptic operators in honeycomb structures

Maxence Cassier

27/06/2024 15:30

In this talk, we analyse the propagation of Transverse Electric (TE) waves in a two dimensional honeycomb photonic medium. This medium consists of an homogeneous bulk of fixed permittivity and an array of high permittivity dielectric inclusions centered at the vertices of a honeycomb lattice. In the high contrast regime, we perform a mathematical study of the band structure of the photonic...

20. Tunneling in radial potential wells and constant magnetic field or Aharonov-Bohm

Bernard Helffer

28/06/2024 09:00

We investigate a Hamiltonian with radial potential wells and either a constant magnetic field or an Aharonov-Bohm magnetic field. Assuming that the potential wells are symmetric, we derive the semi-classical asymptotics of the splitting between the ground and second state energies. The first case was first considered by Fefferman-Shapiro-Weinstein and improved successively by Helffer-Kachmar,...

21. Resolvent estimates for one-dimensional Dirac operators with imaginary potentials

Petr Siegl

28/06/2024 10:30

We consider one-dimensional Dirac operators on the real line with imaginary potentials unbounded at infinity. Lower resolvent norm estimates were obtained in [Nguyen-Krejcirik-22] via a (non-semiclassical) pseudomode construction for the spectral parameter diverging to infinity in various regions of the complex plane. Our results comprise upper resolvent norm estimates in the complementary...

22. Can one hear an obstacle subject to a constant magnetic field?

Vincent Bruneau

28/06/2024 11:30

We consider the Schrödinger and Dirac operators with constant magnetic field in dimensions 2 and 3. The aim of this talk is to give an overview of the known results concerning the distribution of the spectrum of these operators when they are perturbed by an obstacle (operator outside a bounded domain). In the absence of a magnetic field, from Weyl's formulas, it is known that at least the...