The Borg-Marchenko uniqueness theorem for complex potentials

Abstract

We introduce and study a new theoretical concept of \textit{spectral pair} for a Schrödinger operator $H$ in $L^2({\mathbb{R}}_{+})$ with a bounded \textit{complex-valued} potential. The spectral pair consists of a scalar measure and a complex-valued function. We show that in many ways, the spectral pair generalises the classical spectral measure to the non-self-adjoint case. First, extending the classical Borg-Marchenko theorem, we prove a uniqueness result: the spectral pair uniquely determines the operator $H$. Second, we derive asymptotic formulas for the spectral pair in the spirit of the classical result of Marchenko. In the case of real-valued potentials, we relate the spectral pair to the spectral measure of $H$. Lastly, we provide formulas for the spectral pair at a~simple eigenvalue of~$|H|$.

v2: complex Robin boundary parameter; 52 pages