On Lame's equation of a particular kind

TL;DR

This paper demonstrates how a specific form of Lame's equation can be transformed into a hypergeometric equation, providing explicit expressions for characteristic exponents and an analytical condition for parametric amplification.

Contribution

It introduces a reduction of a particular Lame's equation to a hypergeometric form and derives explicit characteristic exponents and amplification conditions.

Findings

Reduction of Lame's equation to hypergeometric form

Explicit expressions for characteristic exponents

Analytical condition for parametric amplification

Abstract

It is shown that the Lame's equation ${d^{2}\over d z^{2}}X+\kappa^{2} cn^{2}(z, {1\over \sqrt 2})X=0$ can be reduced to the hyper-geometric equation. The characteristic exponents of this equation are expressed in terms of elementary functions of the parameter $\kappa$. An analytical condition for parametric amplification is obtained.