A representation of the transmutation kernels for the Schrödinger operator in terms of eigenfunctions and applications
Provides a mathematical tool for solving Sturm-Liouville problems, but the contribution is incremental and domain-specific.
The authors derive representations of transmutation kernels for the 1D Schrödinger operator using eigenfunctions, improve series convergence to uniform and absolute, and provide a numerical illustration.
The representations of the kernels of the transmutation operator and of its inverse relating the one-dimensional Schrödinger operator with the second derivative are obtained in terms of the eigenfunctions of a corresponding Sturm-Liouville problem. Since both series converge slowly and in general only in a certain distributional sense we find a way to improve these expansions and make them convergent uniformly and absolutely by adding and subtracting corresponding terms. A numerical illustration of the obtained results is given.