Calculation of Characteristic Values for Periodic Potentials

Abstract

Section II reviews the general properties of the solutions of differential equations with periodic coefficients, the socalled Hill and Mathieu equations. In Section III asymptotic formulae for the eigenvalues of the Mathieu equation in the oscillatory region (Eq. (9)) are obtained by applying the Schrödinger perturbation theory to the problem of the physical pendulum. Section IV applies the W. K. B. method to the Hill equations. Implicit equations for the eigenvalues are obtained (§9, Eq. (28), (30) §10). They are valid for $E<\mathrm{}{V}_{max}$. In the introduction (Section I) a group of physical problems are mentioned, to which the results may be applied.