Stability in the inverse resonance problem for the Schr\" odinger operator

TL;DR

This paper establishes a quantitative stability estimate for reconstructing a potential in the Schrödinger equation from finitely many resonances, showing the error decreases polynomially as the resonance disk radius increases.

Contribution

It provides the first explicit stability estimate for the inverse resonance problem for Schrödinger operators with finite support potentials.

Findings

Potential can be recovered with accuracy improving as more resonances are used.

The error bound decreases polynomially with the radius of the resonance disk.

Constants depend on prior information about the potential.

Abstract

We work with the Schr\" odinger equation \begin{equation*} H_q y = -y'' + q(x)y = z^2y, \ x\in [0,\infty), \end{equation*} where $q\in L_1((0,\infty), xdx)$, and asssume that the corresponding operator $H_q$ is defined by the Dirihlet condition $y(0) = 0$ The function $\psi(z) = y(0,z)$ where $y(x,z)$ is the Jost solution of the above equation is analytic in the whole complex plane, provided that the support of the potential $q$ is finite. The zeros of $\psi$ are called the resonances. It is known that $q$ is uniquely determined by the sequence of resonances. Using only finitely many resonances lying in the disk $|z|\le r$ we can recover the potential $q$ with accuracy $\varepsilon(r)\to 0$ as $r \to \infty$. The main result of the paper is the estimate $\varepsilon(r) \le Cr^{-\alpha}$ with some constants $C$ and $\alpha>0$ which are defined by a priori information about the potential $q$.