Levinson's theorem for dissipative systems, or how to count the asymptotically disappearing states

We consider dissipative Schroedinger operators of the form $H=-\mathrm{\Delta }+V(x)$ on $L^2({\mathbb{R}}^3)$, with $V(x)$ a complex, bounded and decaying potential with a non-positive imaginary part. We prove a topological version of Levinson’s theorem corresponding to an index theorem for the discrete, complex spectrum of $H$.