A lower bound and estimate of the lowest eigenvalue of a second order Floquet equation

A bound on the lowest eigenvalue of a second order Floquet problem is derived by applying the Cauchy Integral Theorem. Specifically, we chose a special function which depends on an arbitrary positive parameter $S\ge 1$. We use the residue theorem and show that its residue at the origin determines an infinite sum composed of reciprocals of the eigenvalues raised to the $2S$ power. A simple bound gives us our result. We show that the residue depends explicitly on the power series expansions, in the eigen-parameter, of the original equation’s fundamental solutions. The coefficients of these power series are computed recursively.

Three typical examples are presented, and it is shown for these cases that the lower bound, derived in this paper, actually affords a good approximation to the first eigenvalue. We show this for the case only of $S=1$.