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, a∈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 Fa(z;q)=ψx(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 Fa(z;q)=iz−a+O(1) rather than 1+O(z−1). In the transformation-operator argument the relevant boundary quantity is Bx(0,t)−aB(0,t), together with the scalar term B(0,0). For fixed a∈ℂ, and uniformly for a in bounded subsets of 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.
Related
- A spectral-element method for computing many eigenvalues and their asymptotics of the Schrödinger operator with Robin boundary condition
- Inverse spectral results for the magnetic Schrödinger operator in low dimensions
- Optimization Inverse Spectral Problem for the Schrödinger Operator on a Compact Graph
- A Schiffer-type inverse obstacle problem for the planar Schrödinger equation with a single measurement
- On the gauge invariant boundary condition in open XXZ chains: SoV bases, elementary blocks and overlaps
BibTeX
@misc{shterenberg2026inverse,
title = {{Inverse Resonance Problem for the Half-Line Schrödinger Operator with a Complex Robin Boundary Condition}},
author = {Roman Shterenberg},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14323},
eprint = {2610.14323},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14323},
note = {CC0},
}
RIS
TY - UNPB
TI - Inverse Resonance Problem for the Half-Line Schrödinger Operator with a Complex Robin Boundary Condition
AU - Roman Shterenberg
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.14323
UR - https://arxiv.org/abs/2610.14323
AB - 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, a∈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∈ℂ, and uniformly for a in bounded subsets of 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.
ER -