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arXiv · 1504.02003

On Sturm-Liouville Equations with Several Spectral Parameters

R. Michael Porter
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Abstract

We give explicit formulas for a pair of linearly independent solutions of $(py')'(x)+q(x)=(λ_1r_1(x)+\cdots+λ_dr_d(x))y(x)$, thus generalizing to arbitrary $d$ previously known formulas for $d=1$. These are power series in the spectral parameters $λ_1,\dots,λ_d$ (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.

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R. Michael Porter. 2015-04-08. On Sturm-Liouville Equations with Several Spectral Parameters. https://doi.org/10.1007/s40590-015-0078-2

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