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Mesures limites pour l’équation de Helmholtz dans le cas non captif
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 18 (2009) no. 3, pp. 445-479.

This paper is devoted to the study of the limit measures associated with the solution of the Helmholtz equation with a source term which concentrates on a point. The potential is assumed to be C and the operator non trapping. The solution of the semi-classical Schrödinger equation is written micro-locally as a finite sum of Lagrangian distributions. Under a geometrical hypothesis, which generalizes the virial assumption, this representation implies that the limit measure exists and satisfies standard properties. Finally, one gives an example of operator which does not satisfy the geometrical hypothesis and for which the limit measure is not unique. The case of two source terms is also treated

Cet article est consacré à l’étude des mesures limites associées à la solution de l’équation de Helmholtz avec un terme source se concentrant en un point. Le potentiel est supposé C et l’opérateur non-captif. La solution de l’équation de Schrödinger semi-classique s’écrit alors micro-localement comme somme finie de distributions lagrangiennes. Sous une hypothèse géométrique, qui généralise l’hypothèse du viriel, on en déduit que la mesure limite existe et qu’elle vérifie des propriétés standard. Enfin, on donne un exemple d’opérateur qui ne vérifie pas l’hypothèse géométrique et pour lequel la mesure limite n’est pas unique. Le cas de deux termes sources est aussi traité.

DOI: 10.5802/afst.1210
Jean-François Bony 1

1 Institut de Mathématiques de Bordeaux, UMR 5251 du CNRS, Université de Bordeaux I, 351 cours de la Libération, 33405 Talence, France
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     author = {Jean-Fran\c{c}ois Bony},
     title = {Mesures limites pour l{\textquoteright}\'equation de {Helmholtz} dans le cas non captif},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {445--479},
     publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
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Jean-François Bony. Mesures limites pour l’équation de Helmholtz dans le cas non captif. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 18 (2009) no. 3, pp. 445-479. doi : 10.5802/afst.1210. https://afst.centre-mersenne.org/articles/10.5802/afst.1210/

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