# Application of the boundary control method to partial data Borg-Levinson inverse spectral problem

## TL;DR

This paper demonstrates that boundary spectral data on an arbitrary portion of the boundary uniquely determine a bounded potential in a multidimensional Schrödinger operator, extending previous results to less smooth potentials using the Boundary Control method.

## Contribution

It introduces a novel application of the Boundary Control method to prove uniqueness of the potential from partial boundary spectral data for less regular potentials.

## Key findings

- Unique determination of bounded potential from partial boundary spectral data.
- Extension of previous smoothness assumptions to bounded potentials.
- Self-contained presentation of the Boundary Control method for inverse spectral problems.

## Abstract

We consider the multidimensional Borg-Levinson problem of determining a potential $q$, appearing in the Dirichlet realization of the Schr\"odinger operator $A_q=-\Delta+q$ on a bounded domain $\Omega\subset \mathbb{R}^n$, $n\geq2$, from the boundary spectral data of $A_q$ on an arbitrary portion of $\partial\Omega$. More precisely, for $\gamma$ an open and non-empty subset of $\partial\Omega$, we consider the boundary spectral data on $\gamma$ given by $\mathrm{BSD}(q,\gamma):=\{(\lambda_{k},{\partial_\nu \phi_{k}}_{|\overline{\gamma}}):\ k \geq1\}$, where $\{ \lambda_k:\ k \geq1\}$ is the non-decreasing sequence of eigenvalues of $A_q$, $\{ \phi_k:\ k \geq1 \}$ an associated Hilbertian basis of eigenfunctions, and $\nu$ is the unit outward normal vector to $\partial\Omega$. We prove that the data $\mathrm{BSD}(q,\gamma)$ uniquely determine a bounded potential $q\in L^\infty(\Omega)$. Previous uniqueness results, with arbitrarily small $\gamma$, assume that $q$ is smooth. Our approach is based on the Boundary Control method, and we give a self-contained presentation of the method, focusing on the analytic rather than geometric aspects of the method.