A regularity result for the bound states of NN-body Schr\"odinger operators: Blow-ups and Lie manifolds

We prove regularity estimates for the eigenfunctions of Schr\"odinger type operators whose potentials have inverse square singularities and uniform radial limits at infinity. In particular, the usual $N$-body operators are covered by our result; in that case, the weight is in terms of the (euclidean) distance to the collision planes. The technique of proof is based on blow-ups and Lie manifolds. More precisely, we first blow-up the spheres at infinity of the collision planes to obtain the Georgescu-Vasy compactification and then we blow-up the collision planes. We carefully investigate how the Lie manifold structure and the associated data (metric, Sobolev spaces, differential operators) change with each blow-up. Our method applies also to higher order operators and matrices of scalar operators.Comment: 19 page

Analyse sur les espaces singuliers et algèbres d’opérateurs

Nous étudions l'opérateur H=-Δ +V qui représente l'énergie d'un système à N-électrons. Pour cela, nous utilisons les algèbres d'opérateurs. Nous commençons par définir une C*-algèbres A qui contient le potentiel V du problème puis nous prenons son produit croisé AxX . Les résolvantes de H sont ainsi contenues dans cette C*algèbre dans AxX. Par une étude précise du spectre de AxX, nous obtenons une décomposition spectrale essentiel de H et donc un résultat qui étend le théorème HV Z dans la continuité des travaux de V. Georgescu. Nous étendons ce résultat en remplaçant l'espace euclidien X par le groupe de Heisenberg. Dans la seconde partie de la thèse, nous montrons que le spectre de la C*-algèbre A et un espace introduit par A. Vasy dans les années 2000 sont les mêmes. L'espace construit par A. Vasy est construit par éclatements successifs d'une variété différentielle à coins. La preuve repose également sur des résultats d'éclatements de variétés. En particulier, nous avons introduit la notion de « graph blow-up »' d'une variété par rapport à une famille assez générale de sous-variétés.We study the operator H=-Δ +V that describes the energy of a system with N electrons. To do this, we use operator algebras. We thus first define a C*algebra A that contains the potentials V of the problem and then consider the crossed product AxX. The resolvents of H then belong to the C*algebra AxX. By a precise study of the spectrum of AxX, we obtain a decomposition of the essential spectrum of H, and hence of result that extends the HVZ theorem, in the spirit of Georgescu. We extend these results by replacing the underlying Euclidean space X with the Heisenberg group. In the second part of the thesis, we show that the spectrum of $A$ and the space introduce by A. Vasy around the year 2000 are the same. The space introduced by A. Vasy is defined using the blow-up of differentials manifolds with corners. The proofs are based on some differential geometric results on blow-ups of manifolds, in particular, we introduce the notion of ``graph blow-up'' of a manifold with respect to a rather general family of submanifolds

A regularity result for the bound states of N-body Schrödinger operators: blow-ups and Lie manifolds

We prove regularity estimates in weighted Sobolev spaces for the L2-eigenfunctions of Schrödinger-type operators whose potentials have inverse square singularities and uniform radial limits at infinity. In particular, the usual N-body Hamiltonians with Coulomb-type singular potentials are covered by our result: in that case, the weight is , where is the usual Euclidean distance to the union of the set of collision planes F. The proof is based on blow-ups of manifolds with corners and Lie manifolds. More precisely, we start with the radial compactification X¯¯¯¯ of the underlying space X and we first blow up the spheres SY⊂SX at infinity of the collision planes Y∈F to obtain the Georgescu–Vasy compactification. Then, we blow up the collision planes F. We carefully investigate how the Lie manifold structure and the associated data (metric, Sobolev spaces, differential operators) change with each blow-up. Our method applies also to higher-order differential operators, to certain classes of pseudodifferential operators, and to matrices of scalar operators