We continue the analysis of algebras introduced by Georgescu, Nistor, and their coauthors, in order to study N-body type Hamiltonians with interactions. More precisely, let be a linear subspace of a finite-dimensional Euclidean space X, and be a continuous function on that has uniform homogeneous radial limits at infinity. We consider, in this paper, Hamiltonians of the form , where the subspaces belong to some given family of subspaces. Georgescu and Nistor have considered the case when consists of all subspaces , and Nistor and coauthors considered the case when is a finite semilattice and Georgescu generalized these results to any family. In this paper, we develop new techniques to prove their results on the spectral theory of the Hamiltonian to the case where is any family of subspaces also, and extend those results to other operators affiliated to a larger algebra of pseudodifferential operators associated with the action of X introduced by Connes. In addition, we exhibit Fredholm conditions for such elliptic operators. We also note that the algebras we consider answer a question of Melrose and Singer.
Nous poursuivons l'étude des algèbres introduites par Georgescu, Nistor et leurs collaborateurs pour étudier les hamiltoniens du problème à N corps avec interactions. Plus précisément, soit un sous-espace d'un espace euclidien X de dimension finie, et soit une fonction continue sur possédant des limites radiales à l'infini. Nous considérons ici des hamiltoniens de la forme , où est une famille donnée de sous-espaces de X. Georgescu et Nistor ont étudié en détail le cas où la famille contient tous les sous-espaces de X ; Nistor et ses co-auteurs ont étudié le cas d'une famille finie stable par intersections, et Georgescu a généralisé certains de ces résultats à des familles quelconques. Dans cette note, nous développons de nouvelles techniques pour étendre les résultats spectraux précédents à des familles quelconques également et pour des opérateurs pseudo-différentiels (ou affiliés) associés à l'action de X par translation, introduits par Connes. En outre, nous obtenons, pour les opérateurs elliptiques, une caractérisation comme étant des opérateurs de Fredholm. Nous faisons remarquer également que ces algèbres répondent à une question de Melrose et Singer.
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Jérémy Mougel 1; Nicolas Prudhon 1
@article{CRMATH_2019__357_2_200_0, author = {J\'er\'emy Mougel and Nicolas Prudhon}, title = {Exhaustive families of representations of {\protect\emph{C}\protect\textsuperscript{{\textasteriskcentered}}-algebras} associated with {\protect\emph{N}-body} {Hamiltonians} with asymptotically homogeneous interactions}, journal = {Comptes Rendus. Math\'ematique}, pages = {200--204}, publisher = {Elsevier}, volume = {357}, number = {2}, year = {2019}, doi = {10.1016/j.crma.2019.01.010}, language = {en}, }
TY - JOUR AU - Jérémy Mougel AU - Nicolas Prudhon TI - Exhaustive families of representations of C⁎-algebras associated with N-body Hamiltonians with asymptotically homogeneous interactions JO - Comptes Rendus. Mathématique PY - 2019 SP - 200 EP - 204 VL - 357 IS - 2 PB - Elsevier DO - 10.1016/j.crma.2019.01.010 LA - en ID - CRMATH_2019__357_2_200_0 ER -
%0 Journal Article %A Jérémy Mougel %A Nicolas Prudhon %T Exhaustive families of representations of C⁎-algebras associated with N-body Hamiltonians with asymptotically homogeneous interactions %J Comptes Rendus. Mathématique %D 2019 %P 200-204 %V 357 %N 2 %I Elsevier %R 10.1016/j.crma.2019.01.010 %G en %F CRMATH_2019__357_2_200_0
Jérémy Mougel; Nicolas Prudhon. Exhaustive families of representations of C⁎-algebras associated with N-body Hamiltonians with asymptotically homogeneous interactions. Comptes Rendus. Mathématique, Volume 357 (2019) no. 2, pp. 200-204. doi : 10.1016/j.crma.2019.01.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2019.01.010/
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