Théorie Spectrale et Analyse Semiclassique
Université Paris 13, Villetaneuse, 21-23 septembre, 2009Résumés
Vladimir
Buslaev (St Petersburg University)
New approach to the quantum three-body scattering problem. One dimensional charged particles. The talk is based on a joint work with Serguei Levin.
In [1] we described a new approach to the scattering in the system of three one-dimensional quantum particles with short range pair interactions. We could find the singularities in the asymptotic behavior of the eigenfunctions at infinity in the configuration space. These results were used for the numerical computations of scattered plane waves. In the present talk we are going to describe how to generalize this approach to the case of charged particles.
[1] Buslaev, V.S. and Levin, S.B. American Math. Soc. Transl. (2), v. 225 (2008) 55-71
New approach to the quantum three-body scattering problem. One dimensional charged particles. The talk is based on a joint work with Serguei Levin.
In [1] we described a new approach to the scattering in the system of three one-dimensional quantum particles with short range pair interactions. We could find the singularities in the asymptotic behavior of the eigenfunctions at infinity in the configuration space. These results were used for the numerical computations of scattered plane waves. In the present talk we are going to describe how to generalize this approach to the case of charged particles.
[1] Buslaev, V.S. and Levin, S.B. American Math. Soc. Transl. (2), v. 225 (2008) 55-71
Mouez Dimassi
(Université Paris 13)
Quelques propriétes spectrales de l' opérateur de Schrödinger avec champ magnétique en dimension deux Nous donnons quelques résultats concernant les valeurs propres, le résonances et la fonction de décalage spectrale pour l'opérateur de Schrödinger avec champ magnétique et cham électrique exterieurs. Nous discutons deux régimes : champ magnétique fort et champ électrique faible.
Quelques propriétes spectrales de l' opérateur de Schrödinger avec champ magnétique en dimension deux Nous donnons quelques résultats concernant les valeurs propres, le résonances et la fonction de décalage spectrale pour l'opérateur de Schrödinger avec champ magnétique et cham électrique exterieurs. Nous discutons deux régimes : champ magnétique fort et champ électrique faible.
Alexandre
Fedotov (St Petersburg University)
Typical growth of Gaussian sums We study exponential sums containing general quadratic polynomial $an^2+bn+c$ in the exponent ($n$ is the number of the term of the exponential sum). So, there are two parameters that determine the growth of the exponential sums: the frequency $a$ and the ergodic parameter $b$. We show that, for almost all $a$ and $b$, there is a typical growth of the exponential sum, and that there is an exceptional set of values of $b$ where the exponential sums grow anamalously fast. The work is done in collaboration with F.Klopp.
Typical growth of Gaussian sums We study exponential sums containing general quadratic polynomial $an^2+bn+c$ in the exponent ($n$ is the number of the term of the exponential sum). So, there are two parameters that determine the growth of the exponential sums: the frequency $a$ and the ergodic parameter $b$. We show that, for almost all $a$ and $b$, there is a typical growth of the exponential sum, and that there is an exceptional set of values of $b$ where the exponential sums grow anamalously fast. The work is done in collaboration with F.Klopp.
Christian Gérard
(Université Paris Sud 11)
Le modèle de Nelson à coefficients variables. Nous décrirons dans cet exposé quelques résultats préliminaires sur le modèle de Nelson à coefficients variables, qui apparait quand on considère l'équation de Klein-Gordon sur des espaces-temps statiques couplée à l'équation de Newton. Nous discuterons le problème ultraviolet, en montrant que la troncature ultraviolette peut être éliminé comme dans le modèle de Nelson habituel. Nous discuterons aussi le problème infrarouge (existence d'un état fondamental dans l'espace de Hilbert), en montrant que si la masse décroit vers 0 assez lentement, il existe un état fondamental. Enfin nous donnerons quelques problèmes ouverts et quelques conjectures.
Le modèle de Nelson à coefficients variables. Nous décrirons dans cet exposé quelques résultats préliminaires sur le modèle de Nelson à coefficients variables, qui apparait quand on considère l'équation de Klein-Gordon sur des espaces-temps statiques couplée à l'équation de Newton. Nous discuterons le problème ultraviolet, en montrant que la troncature ultraviolette peut être éliminé comme dans le modèle de Nelson habituel. Nous discuterons aussi le problème infrarouge (existence d'un état fondamental dans l'espace de Hilbert), en montrant que si la masse décroit vers 0 assez lentement, il existe un état fondamental. Enfin nous donnerons quelques problèmes ouverts et quelques conjectures.
Bernard Helffer
(Université Paris Sud 11)
Semi-classical analysis for magnetic bottles. In this talk we will review the results devoted to the analysis of the bottom of the spectrum for a semi-classical Schrödinger operator with magnetic potentials (but no electric potential). As observed in the application to superconductivity, the analysis could depend strongly on the boundary condition. We will present some recent results of N. Raymond and of Helffer-Kordyukov, in continuation of previous works done in collaboration with A. Morame and S. Fournais.
Semi-classical analysis for magnetic bottles. In this talk we will review the results devoted to the analysis of the bottom of the spectrum for a semi-classical Schrödinger operator with magnetic potentials (but no electric potential). As observed in the application to superconductivity, the analysis could depend strongly on the boundary condition. We will present some recent results of N. Raymond and of Helffer-Kordyukov, in continuation of previous works done in collaboration with A. Morame and S. Fournais.
Frédéric Hérau
(Université de Reims)
Tunnelling effect for Krammers Fokker-Planck operators, the multiple well case. In this work we give a complete picture of the tunnelling effect for Krammers-Fokker-Planck type operators with supersymmetric structure. The result in the multiple well case is deeply linked with a remarkable PT-type property, and we shall explain some of the main steps of the proof. This is a joint work with J. Sjöstrand and M. Hitrik.
Tunnelling effect for Krammers Fokker-Planck operators, the multiple well case. In this work we give a complete picture of the tunnelling effect for Krammers-Fokker-Planck type operators with supersymmetric structure. The result in the multiple well case is deeply linked with a remarkable PT-type property, and we shall explain some of the main steps of the proof. This is a joint work with J. Sjöstrand and M. Hitrik.
André Martinez
(Università di Bologna)
Analytic singularities for long range Schrödinger equations. This is a joint work with S. Nakamura and V. Sordoni. We consider the Schrödinger equation associated to long range perturbations of the flat Euclidian metric (in particular, potentials growing subquadratically at infinity are allowed). We construct a modified quantum free evolution $G_0(s)$ acting on Sjöstrand's spaces, and we characterize the analytic wave front set of the solution $e^{-itH}u_0$ of the Schrödinger equation, in terms of the semiclassical exponential decay of $G_0(-th^{-1})\T u_0$, where $\T$ stands for the Bargmann-transform. The result is valid for $t<0$ near the forward non trapping points, and for $t>0$ near the backward non trapping points.
Analytic singularities for long range Schrödinger equations. This is a joint work with S. Nakamura and V. Sordoni. We consider the Schrödinger equation associated to long range perturbations of the flat Euclidian metric (in particular, potentials growing subquadratically at infinity are allowed). We construct a modified quantum free evolution $G_0(s)$ acting on Sjöstrand's spaces, and we characterize the analytic wave front set of the solution $e^{-itH}u_0$ of the Schrödinger equation, in terms of the semiclassical exponential decay of $G_0(-th^{-1})\T u_0$, where $\T$ stands for the Bargmann-transform. The result is valid for $t<0$ near the forward non trapping points, and for $t>0$ near the backward non trapping points.
Laurent Michel
(Université de Nice)
Semiclassical analysis of the Metropolis algorithm on bounded domains. We consider the semclassical Metropolis operator on a bounded domain. We obtain a precise description of its spectrum that give useful bounds on rates of convergence for the Metropolis algorithm. As an example, we treat the random placement of N hard discs in the unit square, the original application of the Metropolis algorithm. This is a joint work with P. Diaconis and G. Lebeau.
Semiclassical analysis of the Metropolis algorithm on bounded domains. We consider the semclassical Metropolis operator on a bounded domain. We obtain a precise description of its spectrum that give useful bounds on rates of convergence for the Metropolis algorithm. As an example, we treat the random placement of N hard discs in the unit square, the original application of the Metropolis algorithm. This is a joint work with P. Diaconis and G. Lebeau.
Galina
Perelman (Ecole Polytechnique)
Vey theorem in infinite dimensions and its application to the KdV equation We develop an infinite dimensional version of the Vey theorem and apply it to construct the Birkhoff coordinates for the KdV equation in the vicinity of the origin in $L_0^2(S^1)$. The obtained integrating transformation has the form "identity plus a 1-smoothing map". This is a joint work with S. Kuksin.
Vey theorem in infinite dimensions and its application to the KdV equation We develop an infinite dimensional version of the Vey theorem and apply it to construct the Birkhoff coordinates for the KdV equation in the vicinity of the origin in $L_0^2(S^1)$. The obtained integrating transformation has the form "identity plus a 1-smoothing map". This is a joint work with S. Kuksin.
Thierry Ramond
(Université Paris Sud 11)
Long time behaviour of the Schrödinger group at the barrier top energy This is a joint work with J.-F. Bony, S. Fujiie and M. Zerzeri. We describe the behavior for large times of the semiclassical Schrödinger group, for energies close to the unique global maximum of the potential. We show that for times of order $\ln 1/h$, this behaviour is driven by the so-called barrier top resonances, even though these are quite far from the real axis. This result is based on a precise study of the resolvent in neighborhoods of the resonances.
Long time behaviour of the Schrödinger group at the barrier top energy This is a joint work with J.-F. Bony, S. Fujiie and M. Zerzeri. We describe the behavior for large times of the semiclassical Schrödinger group, for energies close to the unique global maximum of the potential. We show that for times of order $\ln 1/h$, this behaviour is driven by the so-called barrier top resonances, even though these are quite far from the real axis. This result is based on a precise study of the resolvent in neighborhoods of the resonances.
Johannes
Sjöstrand (Université de Dijon)
Distribution of eigenvalues for close to self-adjoint operators in two dimensions. This is joint work with M. Hitrik. In an earlier work with Hitrik and Vu Ngoc we studied the distribution of eigenvalues with imaginary parts close to certain Diophantine levels. In the present work consider a small perturbation of a semi-classical self-adjoint operator whose principal symbol is completely integrable. We get an asymptotic formula for the number of eigenvalues in a band bounded from above and from below by Diophantine levels.
Distribution of eigenvalues for close to self-adjoint operators in two dimensions. This is joint work with M. Hitrik. In an earlier work with Hitrik and Vu Ngoc we studied the distribution of eigenvalues with imaginary parts close to certain Diophantine levels. In the present work consider a small perturbation of a semi-classical self-adjoint operator whose principal symbol is completely integrable. We get an asymptotic formula for the number of eigenvalues in a band bounded from above and from below by Diophantine levels.
Maher Zerzeri
(Université Paris 13)
Fonction de décalage spectrale pour des perturbations de l'opérateur de Schrödinger périodique. C'est un travail en collaboration avec Mouez Dimassi. Nous présenterons dans cette exposé la fonction de décalage spectrale pour différentes perturbations de l'opérateur de Schrödinger périodique, noté $P_0$. Dans la première partie, nous donnerons un développement asymptotique en puissance de $h$ de la dérivée de la fonction de décalage spectrale correspondante au couple $\big(P(h)=P_0+\varphi_0(hy), P_0)$ dans le cas des perturbations lentes. Ici $\varphi_0(y)$ est $C^\infty,$ à valeurs réelles, $\sim^{-n-\epsilon}$ près de l'infini et $h\searrow 0$ un
paramètre semi-classique. Dans la seconde partie, nous nous
intéresserons au cas des grandes constantes de couplage $(\mu\uparrow
\infty)$. Nous obtenons aussi un développement asymptotique en
puissance de $\mu^{-{1\over \delta}}$ de la dérivée de la fonction de
décalage spectrale correspondante au couple $\big(P_\mu=P_0+\mu
W(x),P_0\big),$ o\`u $W(x)$ est positif, satisfaisant $W(x)\sim
w_0({x\over |x|})|x|^{-\delta}$ près de l'infini pour un certain
$\delta>n$ et $w_0\in {C}^\infty(\Bbb S^{n-1};\,\Bbb R_+).$ Ici $\Bbb
S^{n-1}$ est la sphère unité de l'espace $\Bbb R^n$.
Fonction de décalage spectrale pour des perturbations de l'opérateur de Schrödinger périodique. C'est un travail en collaboration avec Mouez Dimassi. Nous présenterons dans cette exposé la fonction de décalage spectrale pour différentes perturbations de l'opérateur de Schrödinger périodique, noté $P_0$. Dans la première partie, nous donnerons un développement asymptotique en puissance de $h$ de la dérivée de la fonction de décalage spectrale correspondante au couple $\big(P(h)=P_0+\varphi_0(hy), P_0)$ dans le cas des perturbations lentes. Ici $\varphi_0(y)$ est $C^\infty,$ à valeurs réelles, $\sim