Liouville-green asymptotic solutions of higher-order differential equations with rapidly oscillating potentials

Abstract EN

The standard Liouville - Green (or WKB) asymptotic approximation to solutions of differential equations is well-known for both second-order and higher-order equations.

This approximation is valid under certain restrictions on the sizes of the derivatives of the coefficients in the equation.

These restrictions imply that the coefficients cannot oscillate too rapidly.

In the second-order case, there is also an asymptotic approximation which is valid in the complementary situation where the coefficients do oscillate rapidly.

The existing proof of this result does not extend to higher-order equations.

In this paper, we develop a new and more powerful method which gives the asymptotic solution of higher-order equations with rapidly oscillating coefficients.