The Borg-Marchenko uniqueness theorem for complex potentials
Alexander Pushnitski, František Štampach
TL;DR
The paper develops a non-self-adjoint inverse spectral theory for half-line Schrödinger operators with bounded complex potentials by introducing a spectral pair (ν, ψ) that plays the role of a two-component spectral descriptor. Through hermitisation to a self-adjoint 2×2 system and a matrix-valued Titchmarsh–Weyl framework, it proves a Borg–Marchenko type uniqueness: the spectral pair uniquely determines both the complex potential q and the boundary parameter α. The work also establishes self-adjoint and normal cases, connects the spectral pair to the classical spectral measure in the real-potential setting via dσ = (1+ψ) dν, and provides high-energy and simple-eigenvalue formulas for the spectral data, including Tauberian-type asymptotics and explicit free-case computations. Together, these results extend inverse spectral theory to non-self-adjoint Schrödinger operators, offering a structurally robust alternative to the spectral measure for analysis and identification of complex potentials and boundary conditions.
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~$|H|$.