WKB methods for difference equations I

Summary

It is shown that the basic principles of the well known WKB methods for differential equations [1, 2] can also be used to investigate other types of linear equations.

Solutions which are basically of exponential character are developed for second order difference equations written in the standard form

$$fn + w + fn - w = 2\sigma (n) fn$$

when σ is a slowly varying function of the independent variable n.

Two approaches are considered. Firstly, using the methods and operators of finite difference calculus, WKB-type solutions are developed in terms of indefinite summations. Secondly, by expanding difference operators in terms of differential operators it is found that difference equations can be treated by extending the framework of the standard WKB method for differential equations. The second approach is found to be of more practical value. For example, it is shown how this method leads to the known Green-Liouville asymptotic expansions [3] for the Bessel functions and Hermite polynomials when applied to the recurrence relations which these functions satisfy.

The solutions break down at “turning points”. General connection formulae are established for linking the WKB-type approximations across such points.