Topological Levinson's theorem for inverse square potentials: complex, infinite, but not exceptional

H Inoue , Serge Richard (1)

1 PSPM - Probabilités, statistique, physique mathématique

In this review paper we carry on our investigations on Schrödinger operators with inverse square potentials on the half-line. Depending on several parameters, such operators possess either a finite number of complex eigen-values, or an infinite one, but also some spectral singularities embedded in the continuous spectrum (exceptional situations). The spectral and the scattering theory for these operators is recalled, and new results for the exceptional cases are provided. Some index theorems in scattering theory are also developed, and explanations why these results can not be extended to the exceptional cases are provided.

Mots clés: Scattering theory; index theorem; spectral singularity; Fredholm; semi-Fredholm

Domaines: Physique mathématique [math-ph]

H Inoue, Serge Richard. Topological Levinson's theorem for inverse square potentials: complex, infinite, but not exceptional. Revue roumaine de mathématiques pures et appliquées, inPress. ⟨hal-02024861⟩