Reconstruction techniques for inverse Sturm-Liouville problems with complex coefficients

Abstract: A variety of inverse Sturm-Liouville problems is considered, including the two-spectrum inverse problem, the problem of recovering the potential from the Weyl function, as well as the recovery from the spectral function. In all cases the potential in the Sturm-Liouville equation is assumed to be complex valued. A unified approach for the approximate solution of the inverse Sturm-Liouville problems is developed, based on Neumann series of Bessel functions (NSBF) representations for solutions and their derivatives. Unlike most existing approaches, it allows one to recover not only the complex-valued potential but also the boundary conditions of the Sturm-Liouville problem. Efficient accuracy control is implemented. The numerical method is direct. It involves only solving linear systems of algebraic equations for the coefficients of the NSBF representations, while eventually the knowledge only of the first NSBF coefficients leads to the recovery of the Sturm-Liouville problem. Numerical efficiency is illustrated by several test examples.