The WKB Method and Differential Consequences of the Riccati Equation

N. L. Chuprikov

Tomsk State Pedagogical University

Abstract: A generalized WKB method based on the use of the differential consequences of the Riccati equation is presented. The method combines the simplicity of the traditional WKB method and the universality of the Maslov method: in the case of a smooth potential with classical turning points in a bounded space interval, the leading term of the expansion is found as a root of an algebraic equation and provides a regular approximate solution in the whole domain of the potential; we can increase the accuracy of this solution by taking new differential consequences into account.

Keywords: WKB method, Riccati equation, differential consequences of the Riccati equation, Maslov method, turning point, Planck constant, Airy function.

UDC: 517.9

Received: 20.07.2009

DOI: 10.4213/mzm8658

 English version: Mathematical Notes, 2011, 89:6, 885–892

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