Sinc method in spectrum completion and inverse Sturm–Liouville problems

Abstract

Cardinal series representations for solutions of the Sturm–Liouville equation $-y^{\prime \prime }+q\left(x\right)y={\rho }^{2}y,x\in (0,L)$ with a complex-valued potential $q\left(x\right)$ are obtained, by using the corresponding transmutation operator. Consequently, partial sums of the series approximate the solutions uniformly with respect to $\rho$ in any strip $\left(\text{Im}\rho \right)<C$ of the complex plane. This property of the obtained series representations leads to their applications in a variety of spectral problems. In particular, we show their applicability to the spectrum completion problem, consisting in computing large sets of the eigenvalues from a reduced finite set of known eigenvalues, without any information on the potential $q\left(x\right)$ as well as on the constants from boundary conditions. Among other applications this leads to an efficient numerical method for computing a Weyl function from two finite sets of the eigenvalues. This possibility is explored in the present work and illustrated by numerical tests. Finally, based on the cardinal series representations obtained, we develop a method for the numerical solution of the inverse two-spectra Sturm–Liouville problem and show its numerical efficiency.