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  • RESUME : Le but de ce projet est l'obtention d'estimations spectrales, microlocales ou semi-classiques pour des opérateurs principalement non-autoajoints, et l'application de ces estimations à des problèmes dynamiques et d'évolution. Cela comprend en particulier des estimations de resolvantes, spectrales, pseudospectrales, numériques, de type loi de Weyl, ou sur les résonances. Les problèmes d'évolution peuvent être de type diffusion, dissipation, amortissement, propagation ou retour à l'équilibre, apparaissant en supraconductivité, en océanographie, en relativité, en théorie cinétique des gaz et plus généralement en physique mathématique. L'idée centrale du projet est de favoriser les interactions entre les chercheurs qui travaillent à l'obtention de ces estimations et ceux qui étudient ou modélisent des problèmes d'évolution.

 

  • ABSTRACT : The aim of this project is to study refined spectral, microlocal or semi-classical estimates for mainly non-selfadjoint operators and their applications to dynamical and evolution problems. This involves in particular resolvent type estimates, spectral and pseudospectral estimates, numerical simulations, Weyl law type estimates and resonances results. By evolution problems we mean scattering, diffusion, dissipation, damping, propagation or return to the equilibrium phenomena, arising in kinetic theory, relativity, superconductivity, oceanography and more generally mathematical physics. The central idea of the project is to help interplay between researchers working on estimates and researchers studying or modeling evolution problems.

 

  • SCIENTIFIC PROGRAM The main goal of this project is to develop sharp microlocal and semiclassical analysis of selfadjoint and non-selfadjoint operators appearing in spectral and evolution problems. The project is split into three main tasks.Each part is based on one major theme. In the first task, the main goal is to study the distribution of eigenvalues in various geometries. This concerns monodromy, perturbed integrable systems (e.g. billiards), manifolds with corners (for magnetic Schrödinger operators), hyperbolic surfaces or even graphs. In these geometries even selfadjoint operators are difficult and deserve attention. However, in some particular perturbative cases, some n.s.a. problems can be tackled (see e.g. the study of non-selfadjoint perturbations of integrable systems, or 2-dimensional problems). Of course in all these geometries, the classical dynamic has a great influence on the results. Typical results concern stability of the spectrum, rigidity, or Weyl-law type estimates. In particular the work on Weyl-law estimates for randomly non-selfadjoint perturbations has to be continued. The second task is devoted to results on resonances and scattering properties and more generally refined spectral results and their natural application to the long time behavior of quantum systems. The first goal of this task is essentially to understand resonances near thresholds for operators magnetic Schrödinger operators or Dirac operators, and also to understand high frequency limits for the Helmholtz equation. In both cases results on the resolvent are crucial and can be obtained in weighted spaces (via complex dilation or Mourre estimates). Viewed in the new spaces the operators are non-selfadjoint. This is the main reason why this type of study is relevant in a n.s.a. oriented project. Resolvent estimates can be used to obtain local decay of energy, and this type of results (high frequencies and thresholds) can also be combined to obtain global resolvent bounds and associated dispersion estimates. We intend also to continue the study of non-selfadjoint perturbation of the Helmholtz equation. A natural application of local resolvent estimates (via Mourre theory or microlocal estimates) is then the study of scattering and inverse scattering properties. Apart from the work on the theory itself we are interested in direct scattering for systems and also inverse scattering in relativity (the aim is in that case to recover the corresponding metric). The third task of this project deals with modeling and studying evolution problems. By modeling we essentially mean reductions to easier characteristic problems. This appears in particular in large scale oceanography field and in superconductivity. As far as oceanography is concerned, reductions of the wave equations to correct scales makes typical behaviors appear (so-called Rossby and Poincaré waves), and they can be described with the help of tools developed in the first task. The main motivation for integrating this subject in NOSEVOL is precisely that deep semi-classical and spectral tools (involving e.g. Bohr-Sommerfeld conditions) are in the core of the description of the phenomenon. Concerning superconductivity, the linearized model of the Ginzburg-Landau equation is non-selfadjoint, as soon as both electric and magnetic fields coexist. And some interesting questions may already arise at the level of n.s.a. characteristic toy models (concerning e.g. the complex Airy operator). Inhomogeneous kinetic equations are intrinsically non-selfadjoint (of type transport + diffusion) and non-linear eigenvalue problems can also be managed according to n.s.a. operators. In particular we intend to study in these last three cases properties of the spectrum near the numerical range and associated pseudospectral estimates, theoretically and numerically. The natural continuation is of course the study of evolution problems. As a particular associated issue, we intend to work on spectral projectors, which are not orthogonal anymore, since the semiclassical description of the evolution may be difficult.

 

  • Eigenvalue problems, geometry and dynamics
    • Dynamics and spectral analysis on manifolds
    • Distribution of eigenvalues and spectral analysis in various geometries
  • Scattering and resonances
    • Helmholtz equation and dissipative quantum scattering
    • Spectral analysis near thresholds
    • Resonances and long time behavior
    • Scattering and inverse scattering theory
  • Modeling, pseudospectrum and evolution problems
    • Large scale oceanography
    • Kinetic theory, Fokker-Planck operators and Witten Laplacian
    • Non-linear eigenvalue problems and simulations
    • Non-selfadjoint problems appearing in superconductivity
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