Modified spectral parameter power series representations for solutions of Sturm-Liouville equations and their applications

Abstract

Spectral parameter power series (SPPS) representations for solutions of Sturm-Liouville equations proved to be an efficient practical tool for solving corresponding spectral and scattering problems. They are based on a computation of recursive integrals, sometimes called formal powers. In this paper new relations between the formal powers are presented which considerably improve and extend the application of the SPPS method. For example, originally the SPPS method at a first step required to construct a nonvanishing (in general, a complex-valued) particular solution corresponding to the zero-value of the spectral parameter. The obtained relations remove this limitation. Additionally, equations with "nasty" Sturm-Liouville coefficients $1/p$ or $r$ can be solved by the SPPS method. We develop the SPPS representations for solutions of Sturm-Liouville equations of the form $$ (p(x)u')'+q(x)u=\sum_{k=1}^N λ^k R_k[u], \quad x\in(a,b)$$ where $R_k[u] :=r_k(x)u+s_k(x)u'$, $k=1,\ldots N$, the complex-valued functions $p$, $q$, $r_k$, $s_k$ are continuous on the finite segment $[a,b]$. Several numerical examples illustrate the efficiency of the method and its wide applicability.