Does Levinson's theorem count complex eigenvalues ?

TL;DR

This paper extends Levinson's theorem to systems with complex eigenvalues, relating eigenvalue count to scattering properties for a Schrödinger operator with inverse square potential.

Contribution

It proposes an extended Levinson's theorem applicable to complex eigenvalues in a specific quantum system involving inverse square potentials.

Findings

Extended Levinson's theorem for complex eigenvalues

Relation between eigenvalues and scattering system winding number

Application to Schrödinger operator with inverse square potential

Abstract

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 Schroedinger 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.