Does Levinson's theorem count complex eigenvalues ?

François Nicoleau (1) , Daniel Parra (2) , Serge Richard (2)

1 LMJL - Laboratoire de Mathématiques Jean Leray
2 PSPM - Probabilités, statistique, physique mathématique

Yes it does ! Indeed an extended version of Levinson's theorem is proposed for a system involving complex eigenvalues. The perturbed system corresponds to a realization of the Schrödinger operator with inverse square potential on the half-line, while the Dirichlet Laplacian on the half-line is chosen for the reference system. The resulting relation is an equality between the number of eigenvalues of the perturbed system and the winding number of the scattering system together with additional operators living at 0-energy and at infinite energy.

Mots clés: Levinson's theorem

Domaines: Physique mathématique [math-ph]; Théorie spectrale [math.SP]

François Nicoleau, Daniel Parra, Serge Richard. Does Levinson's theorem count complex eigenvalues ?. Journal of Mathematical Physics, 2017. ⟨hal-01397410⟩