Stationary scattering theory on manifolds

Ito, Kenichi; Skibsted, Erik

Annales de l'Institut Fourier, Tome 71 (2021) no. 3, pp. 1065-1119.

Sur la base de nos travaux antérieurs, nous développons une théorie stationnaire de la diffusion pour l’opérateur de Schrödinger sur une variété possédant une fonction d’échappement. Une classe particulière d’exemples sont les variétés à extrémités euclidiennes et/ou hyperboliques. La diffusion par des obstacles, éventuellement non lisses et/ou non bornés d’une certaine manière, est incluse dans la théorie. Nous développons la théorie en grande partie selon les idées classiques de Jäger, Saitō et Constantin, et dérivons en particulier les asymptotiques WKB des fonctions propres généralisées minimales. Comme application, nous prouvons une conjecture de Hempel, Post et Weder sur les transmissions transversales sous sa forme naturelle et forte dans le cadre de notre théorie.

Based on our previous work we develop a stationary scattering theory for the Schrödinger operator on a manifold possessing an escape function. A particular class of examples are manifolds with Euclidean and/or hyperbolic ends. Scattering by obstacles, possibly non-smooth and/or unbounded in a certain manner, is included in the theory. We develop the theory largely along the classical lines of Jäger, Saitō and Constantin, and derive in particular WKB-asymptotics of minimal generalized eigenfunctions. As an application we prove a conjecture of Hempel, Post and Weder on cross-ends transmissions in its natural and strong form within the framework of our theory.

Classification : 47A40, 58J50, 81U05
Keywords: scattering theory, Schrödinger operator, manifolds
Mot clés : théorie de la diffusion, l’opérateur de Schrödinger, variétés

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