Inverse Resonance Problem for the Half-Line Schrödinger Operator with a Complex Robin Boundary Condition

Abstract

We study the inverse resonance problem for the one-dimensional Schrödinger operator on the half-line with a compactly supported complex-valued potential and a complex Robin boundary condition \[ u'(0)-a u(0)=0,\qquad a\in\mathbb C. \] The proof follows the transformation-operator, entire-function, and Fourier analytic strategy developed by Marletta, Shterenberg and Weikard for the Dirichlet problem \cite{MSW}. The Dirichlet Jost function $ψ(z,0)$ is replaced by the Robin characteristic function \[ F_a(z;q)=ψ_x(z,0;q)-aψ(z,0;q). \] We establish the corresponding large-energy estimates, Hadamard factorization, finite-data stability, and uniqueness results. The principal new feature is that $F_a(z;q)=iz-a+O(1)$ rather than $1+O(z^{-1})$. In the transformation-operator argument the relevant boundary quantity is $B_x(0,t)-aB(0,t)$, together with the scalar term $B(0,0)$. For fixed $a\in\mathbb C$, and uniformly for $a$ in bounded subsets of $\mathbb C$, the resulting inverse resonance estimates have the same qualitative form as those proved in \cite{MSW} for the Dirichlet problem. In particular, for a fixed Robin parameter $a$, the complete set of Robin eigenvalues and resonances, counted with multiplicity, determines the potential uniquely. The text is produced with the help of AI ChatGPT for Academy.