Spectral parameter power series representation for Hill's discriminant

TL;DR

This paper introduces a spectral parameter power series (SPPS) method to represent Hill's discriminant, demonstrating its invariance under Darboux transformations and its numerical utility in calculating eigenspectra, especially for Mathieu-type equations.

Contribution

The paper develops a novel SPPS-based series representation for Hill's discriminant, extending its applicability and invariance properties, with practical numerical implications.

Findings

Series representation of Hill's discriminant established

Invariance under Darboux transformation demonstrated

Feasibility shown for numerical eigenspectrum calculations

Abstract

We establish a series representation of the Hill discriminant based on the spectral parameter power series (SPPS) recently introduced by V. Kravchenko. We also show the invariance of the Hill discriminant under a Darboux transformation and employing the Mathieu case the feasibility of this type of series for numerical calculations of the eigenspectrum