Spectral Theory and Mathematical Physics
Metz, May 16–18, 2017
Clara Aldana (University of Luxembourg)
Title: Determinants of Laplace operators on surfaces with singularities
Abstract: I will start by defining and motivating the determinant of the Laplace operator on a closed
smooth manifold. Then, following the classical result about the maximization of the determinant of the
Laplace operator on closed surfaces by B. Osgood, R. Phillips and P. Sarnak, I will talk about the corresponding problem on surfaces with singularities. I will consider two settings: surfaces with cusps and
funnels, and surfaces with conical singularities. I will mention some of the results that I have obtained with
my collaborators in these two cases, the technical difficulties that appeared there and how we solved them.
Paulo Carrillo-Rouse (Université Paul Sabatier, Toulouse III)
Title: On Corner’s cycle obstructions for Fredholm boundary conditions for manifolds with corners
Abstract: For a manifold with corners there is a homology theory called conormal homology, defined in
terms of faces and incidences and whose cycles correspond geometrically to corner’s cycles. Its Euler
characteristic, χcn := χ0 − χ1 , is given by the alternated sum of the number of (open) faces of a given
codimension.
In this talk I will report on joint work with Jean-Marie Lescure (Clermont-Ferrand) and explain the
following result: For a compact connected manifold with corners X given as a finite product of manifolds
with corners of codimension less or equal to three we have that if X satisfies the Fredholm Perturbation
property (every elliptic pseudodifferential b-operator on X can be perturbed by a b-regularizing operator
so it becomes Fredholm) then the even Euler corner character of X vanishes, i.e. χ0 (X) = 0.
In many interesting examples from codimension two the even Euler corner character does not vanish.
I will sketch with examples of codimension two some topological formulas that one can obtain for the
obstruction of a given b-elliptic operator to satisfy the Fredholm perturbation property
Christian Gérard (Université Paris-Sud)
Title: Quelques remarques sur l’état de Hartle-Hawking pour des trous noirs éternels
Abstract: Un trou noir éternel est décrit par un espace-temps (M, g) possédant à la fois un champ de
Killing global et un horizon de Killing qui délimite l’intérieur du trou noir. Hartley et Hawking ont conjecturé il y a longtemps l’existence d’un état stationnaire pour un champ de Klein-Gordon quantique sur
M, qui dans l’extérieur du trou noir est un état thermal à température à la température de Hawking TH ,
reliée à la gravité de surface du trou noir. L’existence et les propriétés de l’état de Hartle-Hawking n’ont
été démontrées que récemment par Sanders en 2013. Nous montrerons une manière simple de construire
l’état de Hartle-Hawking et de montrer qu’il vérifie la propriété de Hadamard, en utilisant la rotation de
Wick et la notion du projecteur de Calderon associé à un problème aux limites elliptique.
Nadine Grosse (Universität Freiburg)
Title: Lp -spectrum of the Dirac operator
Abstract: We study the Lp -spectrum of the Dirac operator on complete manifolds. One of the main questions in this context is whether this spectrum depends on p. As a first example where p-independence fails
we compute explicitly the Lp -spectrum for the hyperbolic space and its product with compact spaces. As
an application we give a positive mass theorem for some noncompact spin manifolds that are asymptotic
to products of hyperbolic space with a compact manifold which is useful to understand the behaviour of
Yamabe invariants under surgeries. This is joint work with Bernd Ammann (1405.2830, 1502.05227)
Lisette Jager (Université de Reims Champagne-Ardenne)
Title: Pseudodifferential calculus in an infinite dimensional setting and applications in mathematical
physics
Abstract: This talk presents an analogue of the classical, finite dimensional Weyl calculus on an infinite
dimensional probability space. The Weyl calculus associates, with a convenient function called symbol, a
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pseudodifferential operator. We are concerned here, mainly, with the semiclassical asymptotic expansion
of the symbol of a product of two such operators. We then give an application to mathematical physics.
This talk is based on articles written with L. Amour et J. Nourrigat.
Thierry Jecko (Université de Cergy-Pontoise)
Title: Limiting absorption principle for Schrödinger operators with oscillating potentials.
Abstract: I shall present new results on the continuous spectrum of Schrödinger operators equiped with
a certain class of oscillating potentials. An obstruction to the application of Mourre commutator theory
appears. I shall explain how one can overcome the difficulty. This is a joint work with A. Mbarek.
Yohann Le Floch (Université de Strasbourg)
Title: Joint spectra of semiclassical integrable systems
Abstract: A semiclassical integrable system of rank n is the data of n commuting semiclassical operators
(e.g. h-pseudodifferential or Berezin-Toeplitz operators) whose principal symbols form an integrable system. Much attention has been given recently to the study of the joint spectrum of such a system, and in
particular to the inverse spectral problem. In this talk, I will describe the recent advances and current
questions regarding these topics. This talk will be based on joint works with Alvaro Pelayo (UC San Diego)
and San Vu Ngoc (Rennes 1) and, if time permits, on a work in preparation with Joseph Palmer (Rutgers)
Jean-Marie Lescure (Université Blaise Pascal, Clermont-Ferrand)
Title: Fourier integral operators on Lie groupoids
Abstract: In this talk, I will present the content of two papers (L-Manchon-Vassout: Journal of NCG,
and L-Vassout: arXiv:1601.00932) and also results of a work under progress with Stéphane Vassout.
After reviewing the definition and basic examples of Lie groupoids, I will explain how the convolution
product of distributions on a Lie groupoid G reveals the importance of an associated non trivial Lie
groupoid, namely the cotangent symplectic groupoid T ∗ G discovered by Coste-Dazord-Weinstein.
Secondly, I will describe a suitable subclass of Lagrangian distributions on G, called G-FIOs. Then I
will explain how to develop a calculus for G-FIOs, showing that this essentially boils down to a calculus
on the Lagrangian submanifolds in the cotangent groupoid T ∗ G. In the same spirit, C ∗ -continuity results
will be stated for suitable G-FIOs.
This calculus recovers the Hörmander’s calculus when G is the pair groupoid of a C ∞ -manifold as
well as the Melrose’s calculus when G is the b-groupoid associated with a manifold with boundary, and
produces a framework for Fourier integral operators for any singular manifold having a suitable groupoid
(like, for instance, stratified pseudomanifolds). It also complete the calculus on Lie groups initiated by
Nielsen-Stetkaer.
In the remaining time, I will show that the one parameter group eitP , t ∈ R, where P is an order
1 positive elliptic G-pseudodifferential operator, consists of G-FIOs, and finally explain how this result
could be used in future works to study generalizations of spectral asymptotics for measured foliations.
Based on joint works with Dominique Manchon and Stéphane Vassout.
Konstantin Pankrashkin (Université Paris-Sud)
Title: Self-adjoint operators of the type div sgn ∇
Abstract: Being motivated the study of negative-index metamaterials, we will discuss the definition and
the spectral properties of the operators given by the differential expressions div h ∇ in a bounded domain
U with a function h which is equal to 1 on a part of U and to a constant b < 0 on the rest of U . We will
see how the properties of such operators depend on the parameter b and the geometry of U . In particular, one can have a non-empty essential spectrum. Based on a joint work with Claudio Cacciapuoti and
Andrea Posilicano (University of Insubria).
Angela Pasquale (Université de Lorraine)
Title: Resonances on noncompact Riemannian symmetric spaces and spherical representations
Abstract: We consider the problem of meromorphically extending the resolvent of the Laplacian ∆ of
a Riemannian symmetric space of the noncompact type G/K. When such an extension is possible, its
poles are called the resonances of ∆ and the image of the residue operator at a resonance is a G-module.
The main problems are the existence and the localization of the resonances as well as the study of the
(spherical) representations of G so obtained. In this talk we will describe these representations in some
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low rank situations. This talk is based on joint works with Joachim Hilgert (Universität Paderborn) and
Tomasz Przebinda (University of Oklahoma).
Nicolas Popoff (Université de Bordeaux)
Title: Thresholds in the spectrum of Hamiltonian with translationally invariant magnetic fields
Abstract: The goal of this talk is to present different models with translationally invariant magnetic field,
such as Iwatsuka models. The associated Hamiltonians can be fibered, and the properties of the band
functions can be linked to the propagation properties of quantum states submitted to such magnetic
fields. If the case of states with bounded frequencies is well understood, less is known for the states with
energy at a threshold corresponding to a limit of a band function. Asymptotics of the band functions
for large frequencies allows to have a better description of these states, and is a necessary step for the
description of the resolvent near the thresholds. I will also present in particular cases results concerning
the number of eigenvalues below the essential spectrum, and singularity of the spectral shift functions,
for suitable perturbations of such operators.
Radu Purice (Institute of Mathematics of the Romanian Academy)
Title: Low lying spectral gaps induced by slowly varying magnetic fields
Abstract: Consider a periodic Schrödinger operator in two dimensions, perturbed by a weak magnetic
field whose intensity slowly varies around a positive mean. We show that the bottom of the spectrum of
the corresponding magnetic Schrödinger operator, for non-crossing Bloch eigenvalues, develops spectral
islands separated by gaps, reminding of a Landau-level structure.
Serge Richard (Nagoya University and University of Lyon 1)
Title: Topological Levinson’s theorem: complex and infinite
Abstract: During this seminar we shall present the most recent findings about a topological version of
Levinson’s theorem for systems either with complex eigenvalues or with an infinite number of bound
states. Part of these developments are based on a recent work about Schroedinger operators with inverse
square potential on the half-line.
Itaru Sazaki (Shinshu University)
Title: Embedded Eigenvalue and von Neumann-Wigner Potential for the Relativistic Schrödinger Operator
Abstract: We construct Neumann-Wigner type potentials for the massive relativistic Schrodinger operator in one and three dimensions for which an embedded eigenvalue exists. In the non-relativistic limit
the potentials converge to the classical Neumann-Wigner potential. Hence this result can be considered
as a relativistic generalization of the von Neumann and Wigner’s result.
Akito Suzuki (Shinshu University)
Title: Weak limit theorem for discrete quantum walks
Abstract: Discrete time quantum walks are quantum mechanical counterparts of random walks. The
de Moivre-Laplace theorem (or central limit theorem) give the asymptotic behavior of a random walker.
In this talk, we give a weak limit theorem for a quantum walk with a position-dependent coin, which
describes the asymptotic behavior of the quantum walk.
Katrin Wendland (Universität Freiburg)
Title: On partition functions and their elliptic cousins in conformal field theory
Abstract: In a two-dimensional conformal field theory, the partition function serves as a useful tool which
keeps track of the dimension of the space of states at every energy level. As such, it is closely related to
quantities that are familiar from heat kernel techniques in index theory. Without assuming background
knowledge from conformal field theory, in this talk we will discuss the special properties of these partition
functions and their elliptic cousins.
Robert Yuncken (Université Clermont Auvergne)
Title: A groupoid approach to pseudodifferential operators
Abstract: The tangent groupoid is a geometric device for glueing a pseudodifferential operator to its
principal symbol, via a deformation family. We will discuss a converse to this, which characterizes the
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distributional kernels of PsiDOs in terms of homogeneity with respect to a natural R+-action on the
tangent groupoid. One can see this result as a simple new definition of a classical pseudodifferential operator. Moreover, we will show that, armed with an appropriate generalization of the tangent groupoid,
this approach allows us to easily construct more exotic pseudodifferential calculi, such as the Heisenberg
calculus. (Joint work with Erik van Erp.)
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