On Sturm–Liouville equations with several spectral parameters

Abstract

We give explicit formulas for a pair of linearly independent solutions of $$(py')'(x)+\,q(x)y(x)=(\lambda _1r_1(x)+\cdots +\lambda _dr_d(x))y(x)$$ , thus generalizing to arbitrary d previously known formulas for $$d=1$$ (often referred to as “spectral parameter power series” or “SPPS”). These formulas are power series in the spectral parameters $$\lambda _1,\dots ,\lambda _d$$ (real or complex), with coefficients which are functions on the interval of definition of the differential equation. The coefficients are obtained recursively using indefinite integrals involving the coefficients of lower degree. Examples are provided in which these formulas are used to solve numerically some boundary value problems for $$d=2$$ , as well as an application to transmission and reflectance in optics.