Dispersion relation for energy bands and energy gaps derived by the use of a phase-integral method, with an application to the Mathieu equation

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Abstract
The quantal problem of a particle moving in a one-dimensional periodic potential with one barrier per period is treated rigorously by means of a phase-integral method. A general dispersion formula, relating real energies to real or complex wavenumbers, valid for any conveniently chosen order of the phase-integral approximations used, is derived. The formula allows for the use of modified as well as unmodified phase-integral approximations, and different possibilities are discussed. For instance, a modification introduced by Floyd in the usual first-order JWKB approximation and applicable to the interior of high-energy bands can be utilised in our scheme for the generation of higher-order phase-integral approximations. This approach opens the possibility of using this phase-integral method in situations where the parameters of the problem are such that with unmodified approximations it would not work at all. The accuracy obtainable is illustrated by calculations on the Mathieu potential and comparison with available accurate numerical results.