Relation, expressed in terms of elliptic integrals, for determining characteristic values and characteristic exponents within stable and unstable regions (bands and gaps) associated with the Mathieu potential

Abstract

An approximate relation for the determination of characteristic values and characteristic exponents within stable and unstable regions (bands and gaps), associated with the Mathieu potential, is given. The relation, which is based on the use of the first- or the third-order phase-integral approximation, is expressed in terms of complete elliptic integrals of the first and second kind. The great accuracy of the relation is illustrated numerically.