Spectral and Scattering Theory - MAQM

Jun 24, 2024, 10:00 AM → Jun 28, 2024, 5:55 PM Europe/Paris

Institut de Mathématiques de Toulouse

Jean-Marc Bouclet (Institut de Mathématiques de Toulouse), Julien Royer (Université Paul Sabatier - Toulouse 3)

 

This event is part of the thematic semester Mathematical Aspects of Quantum Mechanics. 

The conference will start on Monday, June 24, at 2 p.m. and finish on Friday, June 28, at 12h30 p.m.

All the talks will take place in building 1R3 at the Université Paul Sabatier (see the map).

Speakers:

• Gregory Berkolaiko (Texas University): Duistermaat index and eigenvalue interlacing for perturbations in boundary conditions
• Sabine Boegli (Durham University): On the discrete eigenvalues of Schrödinger operators with complex potentials
• Vincent Bruneau (Université de Bordeaux): Can one hear an obstacle subject to a constant magnetic field
• Maxence Cassier (Institut Fresnel, Marseille): High contrast elliptic operators in honeycomb structures
• Rayan Fahs (Université Toulouse III)
• Borbala Gerhat (Czech Technical University in Prague): Energy decay for strongly damped wave equations
• Dietrich Häfner (Université Grenoble Alpes): The Unruh state for massless fermions on Kerr spacetime and its Hadamard property
• Bernard Helffer (Nantes Université): Tunneling in radial potential wells and constant magnetic field
• Maxime Ingremeau (Université Côte d'Azur): Scattering resonances of large quantum graphs
• Joachim Kerner (FernUniversität in Hagen): Spectral comparison results on quantum graphs
• Jean Lagacé (King's College London): Spectral decomposition on the space of flat surfaces: Laplacians and Siegel—Veech Transforms
• Stefan Le Coz (Université Toulouse III): Blow-up on a star-graph
• Haruya Mizutani (Osaka University): Uniform resolvent estimates for a few non-elliptic operators
• Alexander Pushnitski (King's College London): Hankel operators with band spectra and elliptic functions
• Constanza Rojas-Molina (CY Cergy Paris Université)
• Petr Siegl (TU Graz): Resolvent estimates for one-dimensional Dirac operators with imaginary potentials
• Gregor Tanner (University of Nottingham): A quantum graph approach to metamaterial design
• Joseph Viola (Nantes Université): Some applications of the (complexified) metaplectic group to Schrödinger evolutions with non-self-adjoint operators
• Brian Winn (Loughborough University): Quantum ergodicity for large quantum graphs

 

Schedule - Abstracts

Poster

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Monday, June 24

1:30 PM → 1:55 PM Welcome

2:00 PM → 2:55 PM Quantum ergodicity for large quantum graphs

Speaker: Brian Winn

3:00 PM → 3:55 PM The Unruh state for massless fermions on Kerr spacetime and its Hadamard property

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. One of the main ingredients of the proof is the scattering theory for the classical Dirac field. The talk is based on joint work with C. Gérard and M. Wrochna (Unruh state), Christiane Klein (case of large angular momentum of the black hole) as well as J.-P. Nicolas (classical scattering theory).

Speaker: Dietrich Häfner

4:00 PM → 4:25 PM Coffee break

4:30 PM → 5:25 PM Transition in the Integrated Density of States of a correlated random Schrödinger operator

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 the strength of the disorder, that is linked to the strength of the reinforcement parameter in random walks. This behavior is in stark contrast with the usual behavior of the IDS in disordered systems, known as Lifshitz tails, which are usually associated to Anderson Localization and pure point spectrum in the random operator. This is joint work with M. Disertori, X. Zeng, and V. Rapenne.

Speaker: Constanza Rojas-Molina

Tuesday, June 25

9:00 AM → 9:55 AM Hankel operators with band spectra and elliptic functions

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 operators. As a consequence, operators H have band spectra (the spectrum of H is the union of disjoint intervals). A striking feature of this model is that flat bands (i.e. intervals degenerating into points, which are eigenvalues of infinite multiplicity) may co-exist with non-flat bands; I will discuss some simple explicit examples of this nature. Key to the spectral analysis of this class of Hankel operator is the theory of elliptic functions; I will explain this connection. This is joint work with Alexander Sobolev (University College London).

Speaker: Alexander Pushnitski

10:00 AM → 10:25 AM Coffee break

10:30 AM → 11:25 AM Spectral analysis of the magnetic Robin Laplacian

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 investigate the contribution of the magnetic field to the lowest eigenvalue asymptotics.

In the case of a large intensity magnetic field, in the semi-classical limit, I will explain how to get a uniform description of the spectrum located between the Landau levels. The corresponding eigenfunctions, called edge states, are exponentially localized near the boundary. By means of a microlocal dimensional reduction, I will explain how to derive a very precise Weyl law and a proof of quantum magnetic oscillations for excited states, and also how to refine simultaneously old results about the low-lying eigenvalues in the Robin case and recent ones about edge states in the Dirichlet case.

Speaker: Rayan Fahs

11:30 AM → 12:25 PM Spectral decomposition on the space of flat surfaces: Laplacians and Siegel—Veech Transforms

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 spectrum. This result is relevant in the geometry of numbers and in dynamics because the modular surface can parameterise the space of all unimodular lattices (and, thus, also the space of all unit area flat tori).

In this talk, I will explain how to extend these ideas to the study of spaces of flat surfaces of higher genus with singularities. We replace the Eisenstein series with the range of the Siegel—Veech transform and in some specific cases can also identify precisely the cusp forms. I will focus on the case of marked flat tori, this space corresponding to the space of affine lattices. In this situation, we can also identify an operator; which is not the Laplacian but a foliated Laplacian; where the natural decomposition corresponds to its spectrum.

This is joint work with Jayadev S. Athreya (Washington), Martin Möller (Frankfurt) and Martin Raum (Chalmers)

Speaker: Jean Lagacé

12:30 PM → 2:20 PM Lunch

2:30 PM → 3:25 PM Uniform resolvent estimates for a few non-elliptic operators

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 Heisenberg group. In both cases, the main technical tool is the method of weakly conjugate operators, which is a variant of Mourre’s commutator method, with conjugate operators involving the generator of dilation and some non-local Sobolev weights. This talk is partly based on joint work with Luca Fanelli, Luz Roncal and Nico Michele Schiavone.

Speaker: Haruya Mizutani

3:30 PM → 4:25 PM Blow-up on a star-graph

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 characterized explicitly. This is a joint work with François Genoud and Julien Royer.

Speaker: Stefan Le Coz

4:30 PM → 4:55 PM Coffee break

Wednesday, June 26

9:00 AM → 9:55 AM Duistermaat index and eigenvalue interlacing for perturbations in boundary conditions

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 interlacing" (for principal submatrices of Hermitian matrices), "Dirichlet-Neumann bracketing" and so on.

We discuss the extension of this idea to general "perturbations in boundary conditions", encoded as interlacing between eigenvalues of two self-adjoint extensions of a fixed symmetric operator with finite (and equal) defect numbers. In this context, even the terms such as "signature of the perturbation" are not immediately clear, since one cannot take the difference of two operators with different domains. However, it turns out that definitive answers can be obtained, and they are expressed most concisely in terms of the Duistermaat index, an integer-valued topological invariant describing the relative position of three Lagrangian planes in a symplectic space. Two of the planes describe the two self-adjoint extensions being compared, while the third one corresponds to the distinguished Friedrichs extension.

We will illustrate our general results with simple examples, avoiding technicalities as much as possible and giving intuitive explanations of the Duistermaat index, the rank and signature of the perturbation in the self-adjoint extension, and the curious role of the third
extension (Friedrichs) appearing in the answers.

Based on a work in progress with Graham Cox, Yuri Latushkin and Selim Sukhtaiev.

Speaker: Gregory Berkolaiko

10:00 AM → 10:25 AM Coffee break

10:30 AM → 11:25 AM Scattering resonances of large quantum graphs

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 graph is large.

Speaker: Maxime Ingremeau

11:30 AM → 12:25 PM Spectral comparison results on quantum graphs

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 different Schrödinger operators defined
on a given metric graph in a suitable way. Establishing so-called local Weyl laws - which
prove interesting in their own right - we shall derive an explicit expression for the limiting
mean-value of eigenvalue distances. We will first look at finite compact metric graphs
and then move on to a certain class of infinite metric graphs. As we will see, some things
might change in the infinite setting. Furthermore, we shall discuss some application of
the results regarding inverse spectral theory. Namely, we derive some seemingly novel
Ambarzumian-type theorems on graphs. This talk is based on recent work with Patrizio
Bifulco (Hagen).

Speaker: Joachim Kerner

12:30 PM → 2:20 PM Lunch

2:20 PM → 6:00 PM Free afternoon

Thursday, June 27

9:00 AM → 9:55 AM On the discrete eigenvalues of Schrödinger operators with complex potentials

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 proves optimality (in some sense) of generalisations of Lieb-Thirring inequalities to the non-selfadjoint case - thus giving us information about the accumulation rate of the discrete eigenvalues to the essential spectrum.
This talk is based on joint works with Jean-Claude Cuenin (Loughborough) and Frantisek Stampach (Prague).

Speaker: Sabine Boegli

10:00 AM → 10:25 AM Coffee break

10:30 AM → 11:25 AM Energy decay for strongly damped wave equations

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 the semigroup. This generalises a result by R. Ikehata and H. Takeda which was obtained by a different approach based on PDE analysis methods.
Based on joint work with A. Arnal, J. Royer and P. Siegl.

Speaker: Borbala Gerhat

11:30 AM → 12:25 PM Some applications of the (complexified) metaplectic group to Schrödinger evolutions with non-self-adjoint operators

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.

Speaker: Joseph Viola

12:30 PM → 2:20 PM Lunch

2:30 PM → 3:25 PM A quantum graph approach to metamaterial design

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 the edges with vertices acting as scatterers modelling sub-wavelength resonant elements. These resonant elements are constructed with the help of finite quantum graphs attached to each vertex of the lattice with customisable properties controlled by a unitary scattering matrix. The metamaterial properties are understood and engineered by manipulating the band diagram of the periodic structure. The engineered properties are then demonstrated in terms of the reflection and transmission behaviour of Gaussian beam solutions at interfaces between different metamaterials. We demonstrate negative refraction, beam steering and wave vector filtering among other effects both within our model and in experiments.

Speaker: Gregor Tanner

3:30 PM → 4:25 PM High contrast elliptic operators in honeycomb structures

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 crystal. Using a combination of rigorous analytical methods, supported by numerical simulations, we obtain detailed local information about the conical crossings of dispersion surfaces (Dirac points) as well as global information about the high contrast behavior of dispersion surfaces.
The results presented here are based on the article [1] and are summarized in the conference paper [2]. In particular, we prove under a non-degeneracy condition (verified numerically) that the first two dispersion surfaces touch at conical singularities, called Dirac points, over the vertices of the Brillouin zone. We also provide asymptotic expansion with respect to the contrast parameter of the Bloch eigenelements associated to the ``Dirac energy''.
For the particular case of circular inclusions, we prove that all these properties hold for an infinite number of bands arbitrarily high in the spectrum. Finally, we will contrast the E&M setting [1] and the quantum model of graphene analyzed in [3].

Joint work with Michael I. Weisntein (Dept. of Applied Physics & Applied Mathematics, and Dept. of Mathematics, Columbia University, New-York, United States).

M. Cassier and M. I. Weinstein were supported in part by Simons Foundation Math + X Investigator Award #376319. M. I. Weinstein was also supported by US National Science Foundation grants DMS-1412560, DMS-1620418 and DMS-1908657.

\noindent [1] M. Cassier and M. I. Weinstein, High contrast elliptic operators in honeycomb structures, Multiscale Modeling & Simulation 19 (4), 1784-1856, 2021, available on Arxiv (https://arxiv.org/abs/2103.16682).

\noindent [2] M. Cassier and M.I. Weinstein, TE Band Structure for High Contrast Honeycomb Media, 2020 Fourteenth International Congress on Artificial Materials for Novel Wave Phenomena (Metamaterials), IEEE, 2020. p. 479-481.

\noindent [3] C.L. Fefferman, J.P. Lee-Thorp, and M.I. Weinstein, ``Honeycomb Schrödinger operators in the strong binding regime '', Comm. in Pure and Appl. Math., vol. 71 (6), p. 178, 2018.

Speaker: Maxence Cassier

4:30 PM → 4:55 PM Coffee break

8:00 PM → 11:00 PM Social Dinner

Friday, June 28

9:00 AM → 9:55 AM Tunneling in radial potential wells and constant magnetic field or Aharonov-Bohm

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, Helffer-Kachmar-Sundqvist, L. Morin. The second case is a work in progress with A. Kachmar.

(after Helffer-Kachmar, Morin...)

Speaker: Bernard Helffer

10:00 AM → 10:25 AM Coffee break

10:30 AM → 11:25 AM Resolvent estimates for one-dimensional Dirac operators with imaginary potentials

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 regions; thereby establishing also an optimality of pseudospectral regions in [Nguyen-Krejcirik-22]. Our proofs are based in particular on a detailed analysis of the Airy-Dirac operator for which the precise asymptotic behavior of the resolvent norm is found.

The talk is based on a joint work with A. Arnal (Graz) and T. D. Nguyen (Prague).

Speaker: Petr Siegl

11:30 AM → 12:25 PM Can one hear an obstacle subject to a constant magnetic field?

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 volume of the obstacle appears in the asymptotic expansions of the Spectral Shift Function (or the Scattering Phase), but in the presence of a constant magnetic field, the spectral structure is different. Spectral asymptotics involve the logarithmic capacity of the obstacle (or its projected in the direction of the magnetic field). Some of these results are in collaboration with G. Raikov and with P. Miranda.

Speaker: Vincent Bruneau