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

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 $q$.