Hybrid Taylor-WKB series

TL;DR

This paper introduces a generalized WKB method called the Hybrid Taylor-WKB series, which incorporates derivatives of the Riccati equation to improve the regularity and accuracy of solutions near potential barriers and turning points.

Contribution

The paper develops a new WKB series formalism that uses Riccati equation derivatives to enhance solution regularity and accuracy across potential barriers.

Findings

Zeroth-order term is regular across the entire potential barrier interval.

Using more Riccati-equation derivatives increases the solution's accuracy.

The method effectively handles turning points in quantum potential problems.

Abstract

A generalized WKB approach for constructing WKB series endowed with some properties of Taylor ones is presented. Apart from the Riccati equation itself its formalism involves also the Riccati-equation's derivatives (REDs) obtained by differentiating of the former with respect to a spatial variable. For any smooth potential barrier given in the finite spatial interval to include turning points, the zeroth-order term of presented WKB series is regular everywhere in this interval. Moreover, the more REDs are used, the more exact the zeroth-order solution is.