Reconstruction techniques for complex potentials

An approach for solving a variety of inverse coefficient problems for the Sturm-Liouville equation -y''+q(x)y={\lambda}y with a complex valued potential q(x) is presented. It is based on Neumann series of Bessel functions representations for solutions. With their aid the problem is reduced to a system of linear algebraic equations for the coefficients of the representations. The potential is recovered from an arithmetic combination of the first two coefficients. Special cases of the considered problems include the recovery of the potential from a Weyl function, the inverse two-spectra Sturm-Liouville problem as well as the recovery of the potential from the output boundary values of an incident wave interacting with the potential. The approach leads to efficient numerical algorithms for solving coefficient inverse problems. Numerical efficiency is illustrated by several examples.