Quantum mechanical inverse problem on a finite interval as an initial-value problem

Abstract

A new approach is proposed to the problem of the reconstruction of the interaction potential on a finite interval from energy levels En and normalization constants Cn, n=1,2, ., infinity . It is based on the investigation of energy levels En(r) of an auxiliary problem on some interval with variable boundary r. It is shown that functions En(r) satisfy the infinite nonlinear system of second-order differential equations. Numbers En, Cn serve as initial conditions for this system: En=En(0), Cn=E'n(0), and functions En(r) are calculated as solutions of an initial-value problem. The interaction potential U(r) is expressed in terms of En(r). Thus the inverse problem is reduced in fact to the initial-value problem. Since En(r) is a monotonic function of r, the presented approach gives some advantages in practical calculations Some numerical examples show that the interaction potential may be reconstructed in this way with high accuracy.