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

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 $$u^{\mathrm{\prime }}(0)-au(0)=0,a\in \mathbb{C}\mathrm{.}$$ 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 $\psi (z,0)$ is replaced by the Robin characteristic function $$F_a(z;q)={\psi }_x(z,0;q)-a\psi (z,0;q)\mathrm{.}$$ 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.