Journées Équations aux dérivées partielles
Nantes, 5-9 juin
GDR 1151 (CNRS)

Ses participants

Résumés / Abstracts

I. Jean-Marie Barbaroux, François Germinet, Serguei Tcheremchantsev (Université de Nantes)

Quantum diffusion and generalized Rényi dimensions of spectral measures.

We estimate the spreading of the solution of the Schrödinger equation asymptotically in time, in term of the fractal properties of the associated spectral measures. For this, we exhibit a lower bound for the moments of order p at time T for the state $\psi$ defined by

$$\frac{1}{T}\int_0^T \Vert \vert X\vert^{p/2} {\rm e}^{-itH}\psi\Vert^2\, {\rm d}t\ .$$

We show that this lower bound can be expressed in term of the generalized Rényi dimension of the spectral measure $\mu_\psi$ associated to the Hamiltonian H and the state $\psi$. We especially concentrate on continuous models.

II. T. Christiansen, M. S. Joshi (University of Missouri)

Recovering Asymptotics at Infinity of Perturbations of Stratified Media.

We consider perturbations of a stratified medium ${\bf R}^{n-1}_x\times {\bf R}_y$, where the operator studied is $c^2(x,y) \Delta$. The function c is a perturbation of c₀(y), which is constant for sufficiently large |y| and satisfies some other conditions. Under certain restrictions on the perturbation c, we give results on the Fourier integral operator structure of the scattering matrix. Moreover, we show that we can recover the asymptotic expansion at infinity of c from knowledge of c₀ and the singularities of the scattering matrix at fixed energy.

III. Piotr T. Chrusciel, Olivier Lengard (Université de Tours)

Polyhomogeneous solutions of wave equations in the radiation regime.

While the physical properties of the gravitational field in the radiation regime are reasonably well understood, several mathematical questions remain unanswered. The question here is that of existence and properties of gravitational fields with asymptotic behavior compatible with existence of gravitational radiation. A framework to study those questions has been proposed by R. Penrose¹, and developed by H. Friedrich ²³⁴ using conformal completions techniques. In this conformal approach one has to 1) construct initial data, which satisfy the general relativistic constraint equations, with appropriate behavior near the conformal boundary, and 2) show a local (and perhaps also a global) existence theorem for the associated evolution problem. In this context solutions of the constraint equations can be found by solving a nonlinear elliptic system of equations, one of which resembles the Yamabe equation (and coincides with this equation in some cases), with the system degenerating near the conformal boundary. In the first part of the talk I (PTC) will describe the existence and boundary regularity results about this system obtained some years ago in collaboration with Helmut Friedrich and Lars Andersson. Some new applications of those techniques are also presented. In the second part of the talk I will describe some new results, obtained in collaboration with Olivier Lengard, concerning the evolution problem.

IV. Søren Fournais (Aarhus University)

On the current of large atoms in strong magnetic fields.

In this talk I will discuss recent results on the magnetisation/current of large atoms in strong magnetic fields. It is known from the work¹² of Lieb, Solovej and Yngvason that the energy and density of atoms in strong magnetic fields are given to highest order by a Magnetic Thomas Fermi theory (MTF-theory) when the magnetic field strength B and nuclear charge Z satisfy $BZ^{-3} \rightarrow 0$. It is, however, equally interesting to establish whether MTF-theory also gives the right asymptotic current. In this talk we will prove that this is indeed the case, at least for moderate magnetic fields. However, we will also prove that approximate ground states do not in general give the right asymptotics for the current.

V. Frédéric Hérau (Université de Nantes)

Une inégalité de Gårding à bord.

The aim of this work is to give a Gårding inequality for pseudodifferential operators acting on functions in $L^2(\mathbb{R} ^n)$ supported in a closed regular region $F \subset \mathbb{R} ^n$. A natural idea is to suppose that the symbol is non-negative in $F \times \mathbb{R} ^n$. Assuming this, we show that this result is true for pseudo-differential operators of order one, when F is the half-space, and under a supplementary weak hypothesis of degeneracy of the symbol on the boundary.

VI. Marc Herzlich (Université de Montpellier II)

Refined Kato inequalities in Riemannian Geometry.

We describe the recent joint work of the author with David M. J. Calderbank and Paul Gauduchon on refined Kato inequalities for sections of vector bundles living in the kernel of natural first-order elliptic operators.

VII. R. G. Froese, Peter D. Hislop (University of Kentucky)

On the distribution of resonances for some asymptotically hyperbolic manifolds.

We establish a sharp upper bound for the resonance counting function for a class of asymptotically hyperbolic manifolds in arbitrary dimension, including convex, cocompact hyperbolic manifolds in two dimensions. The proof is based on the construction of a suitable paramatrix for the absolute S-matrix that is unitary for real values of the energy. This paramatrix is the S-matrix for a model Laplacian corresponding to a separable metric near infinity. The proof of the upper bound on the resonance counting function requires estimates on the growth of the relative scattering phase, and singular values of a family of integral operators.

VIII. Kunihiko Kajitani (Université de Tsukuba)

Propagation of analyticity of solutions to the Cauchy problem for Kirhhoff type equations.

We shall give the local in time existence of the solutions in Gevrey classes to the Cauchy problem for Kirhhoff equations of p-Laplacian type and investigate the propagation of analyticity of solutions for real analytic deta. When p=2, his equation as the global real analytic solution for the real analytic initial data.

IX. Bernard Kay (University of York)

Application of linear hyperbolic PDE to linear quantum fields in curved spacetimes: especially black holes, time machines and a new semi-local vacuum concept.

Several situations of physical importance may be modelled by linear quantum fields propagating in fixed spacetime-dependent classical background fields. For example, the quantum Dirac field in a strong and/or time-dependent external electromagnetic field accounts for the creation of electron-positron pairs out of the vacuum. Also, the theory of linear quantum fields propagating on a given background curved spacetime is the appropriate framework for the derivation of black-hole evaporation (Hawking effect) and for studying the question whether or not it is possible in principle to manufacture a time-machine. It is a well-established metatheorem that any question concerning such a linear quantum field may be reduced to a definite question concerning the corresponding classical field theory (i.e. linear hyperbolic PDE with non-constant coefficients describing the background in question) - albeit not necessarily a question which would have arisen naturally in a purely classical context. The focus in this talk will be on the covariant Klein-Gordon equation in a fixed curved background, although we shall draw on analogies with other background field problems and with the time-dependent harmonic oscillator. The aim is to give a sketch-impression of the whole subject of Quantum Field Theory in Curved Spacetime, focussing on work with which the author has been personally involved, and also to mention some ideas and work-in-progress by the author and collaborators towards a new ``semi-local'' vacuum construction for this subject. A further aim is to introduce, and set into context, some recent advances in our understanding of the general structure of quantum fields in curved spacetimes which rely on classical results from microlocal analysis.

X. Matthias Lesch (University of Arizona)

Essential self-adjointness of symmetric linear relations associated to first order systems.

The purpose of this note is to present several criteria for essential self-adjointness. The method is based on ideas due to Shubin. This note is divided into two parts. The first part deals with symmetric first order systems on the line in the most general setting. Such a symmetric first order system of differential equations gives rise naturally to a symmetric linear relation in a Hilbert space. In this case even regularity is nontrivial. We will announce a regularity result and discuss criteria for essential self-adjointness of such systems. A byproduct of the regularity result is a short proof of a result due to Kogan and Rofe-Beketov⁵: the so-called formal deficiency indices of a symmetric first order system are locally constant on $\mathbb{C}\setminus\mathbb{R}$. The regularity and its corollary are based on joint work with Mark Malamud. Details will be published elsewhere. In the second part we consider a complete Riemannian manifold, M, and a first order differential operator, $D:C_0^\infty(E)\to C_0^\infty(F)$, acting between sections of the hermitian vector bundles E,F. Moreover, let $V:C^\infty(E)\to L^{\infty}_\mathrm{loc}(E)$ be a self-adjoint zero order differential operator. We give a sufficient condition for the Schrödinger operator H=D^tD+V to be essentially self-adjoint. This generalizes recent work of I. Oleinik⁶⁷⁸, M.  Shubin⁹¹⁰, and M.  Braverman¹¹. We essentially use the method of Shubin. Our presentation shows that there is a close link between Shubin's self-adjointness condition for the Schrödinger operator and Chernoff's self-adjointness condition for powers of first order operators. We also discuss non-elliptic operators. However, in this case we need an additional assumption. We conjecture that the additional assumption turns out to be obsolete in general. The criteria we are going to present in the first and second part of this note are very closely related. In fact, after we had done the second part, we saw that the theory can be extended to symmetric linear relations associated to symmetric first order systems.

XI. Eric Carlen, M. C. Carvalho, Michael Loss (Georgia Institute of Technology)

Many-Body Aspects of Approach to Equilibrium.

Kinetic theory and approach to equilibrium is usually studied in the realm of the Boltzmann equation. With a few notable exceptions not much is known about the solutions of this equation and about its derivation from fundamental principles. In 1956 Mark Kac introduced a probabilistic model of N interacting particles. The velocity distribution is governed by a Markov semi group and the evolution of its single particle marginals is governed (in the infinite particle limit) by a caricature of the spatially homogeneous Boltzmann equation. In joint work with Eric Carlen and Maria Carvalho we compute the gap of the generator of this Markov semigroup and show that the best possible rate of approach to equilibrium in the Kac model is precisely the one predicted by the linearized Boltzmann equation. Similar, but less precise results hold for Maxwellian molecules.

XII. Éric Paturel (Université Paris IX)

Solutions of the Dirac-Fock equations without projector.

In this paper we prove the existence of infinitely many solutions of the Dirac-Fock equations with N electrons turning around a nucleus of atomic charge Z, satisfying N < Z+1 and $\alpha \max(Z, N) < 2/(2/\pi + \pi/2)$, where $\alpha$ is the fundamental constant of the electromagnetic interaction (approximately 1/137). This work is an improvement of an article of Esteban-Séré, where the same result was proved under more restrictive assumptions on N.

XIII. Bernard Helffer, Thierry Ramond (Université Paris XI)

Semiclassical expansion for the thermodynamic limit of the ground state energy of Kac's operator.

We continue the study started by the first author of the semiclassical Kac Operator. This kind of operator has been obtained for example by M. Kac as he was studying a 2D spin lattice by the so-called "transfer operator method". We are interested here in the thermodynamical limit $\Lambda(h)$ of the ground state energy of this operator. For Kac's spin model, $\Lambda(h)$ is the free energy per spin, and the semiclassical regime corresponds to the mean-field approximation. Under suitable assumptions, which are satisfied by the physical examples we have in mind, we construct a formal asymptotic expansion for $\Lambda(h)$ in powers of h, from which we derive precise estimates on $\Lambda(h)$. We work in the settings of Standard Functions introduced by J. Sjöstrand for the study of similar questions in the case of Schrödinger operators.

XIV. Alan D. Rendall (MPI für Gravitationsphysik)

Blow-up for solutions of hyperbolic PDE and spacetime singularities..

An important question in mathematical relativity theory is that of the nature of spacetime singularities. The equations of general relativity, the Einstein equations, are essentially hyperbolic in nature and the study of spacetime singularities is naturally related to blow-up phenomena for nonlinear hyperbolic systems. These connections are explained and recent progress in applying the theory of hyperbolic equations in this field is presented. A direction which has turned out to be fruitful is that of constructing large families of solutions of the Einstein equations with singularities of a simple type by solving singular hyperbolic systems. Heuristic considerations indicate, however, that the generic case will be much more complicated and require different techniques.

XV. Fulvio Ricci (Politecnico di Torino)

Solvability of second-order left-invariant differential operators on the Heisenberg group.

We present some recent results, obtained jointly with Detlef Müller, on solvability of operators of the form

$$\sum_{j,k=1}^{2n}a_{jk}V_jV_k+i\alpha U$$

where the V_j are left-invariant vector fields on the Heisenberg group, such that [V_j,V_j+n]=U ( $1\le j\le n$) are the only nontrivial relations, and A=(a_jk) is a complex symmetric matrix with semi-definite real part. The presentation also contains references on the work done in the past few years in this area.

XVI. Johannes Sjöstrand (École Polytechnique)

Asymptotic distribution of eigenfrequencies for damped wave equations.

The eigenfrequencies associated to a damped wave equation, are known to belong to a band parallel to the real axis. We review Weyl asymptotics for the distribution of the real parts of the eigenfrequencies, we show that up to a set of density 0, the eigenfrequencies are confined to a band determined by the Birkhoff limits of the damping coefficient. We also show that certain averages of the imaginary parts converge to the average of the damping coefficient.

XVII. Leonid Parnovski, Alexander V. Sobolev (University of Sussex)

On the Bethe-Sommerfeld conjecture.

We consider the operator in $\mathbb{R} ^d, d\ge 2$, of the form $H = (-\Delta)^l+V, l > 0$ with a function V periodic with respect to a lattice in $\mathbb{R} ^d$. We prove that the number of gaps in the spectrum of H is finite if 8l > d+3. Previously the finiteness of the number of gaps was known for 4l > d+1. Various approaches to this problem are discussed.

XVIII. Tatiana Suslina (Saint Petersbourg)

Absolute continuity of the spectrum of periodic operators of mathematical physics.

The lecture is devoted to the problem of absolute continuity of the spectrum of periodic operators. A general approach to this problem was suggested by L. Thomas in 1973 for the case of the Schrödinger operator with periodic electric potential. Further application of his method to concrete operators of mathematical physics met analytic difficulties. In recent years several new problems in this area have been solved. We propose a survey of known results in this area, including very recent, and formulate unsolved problems.

XIX. Nikolay Tzvetkov (Université de Paris XI)

Bilinear estimates related to the KP equations.

We survey some recent results for the KP-II equation. We also give an idea for treating the ``bad frequency interactions'' of the bilinear estimates in the Fourier transform restriction spaces related to the KP-I equation.

XX. Ari Laptev, Timo Weidl (Royal Institute of Technology)

Recent Results on Lieb-Thirring inequalities.

We give a survey of results on the Lieb-Thirring inequalities for the eigenvalue moments of Schrödinger operators. In particular, we discuss the optimal values of the constants therein for higher dimensions. We elaborate on certain generalisations and some open problems as well.

Édition numérique :

©

Cellule MathDoc (UMS 5638, UJF/CNRS)

©

Laboratoire de Mathématiques (UMR 6629, Université de Nantes)

Édition imprimée : ISBN 2-86939-157-9. Diffusion CNRS ÉDITIONS