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

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 $q\in L_1((0,\infty), xdx)$,andasssumethatthecorrespondingoperator$H_q$ 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 $\mathrm{\mid }z\mathrm{\mid }\le r$ we can recover the potential $q$ with accuracy $\epsilon (r)\to 0$ as $r\to \mathrm{\infty }$. The main result of the paper is the estimate $\varepsilon® \le Cr^{-\alpha}$withsomeconstants$C$and$\alpha>0$ which are defined by a priori information about the potential $q$.