Overlooked Degree of Freedom in Steepest Descent Method: Steepest Descent Method Corresponding to Divergence-Free WKB Method

Abstract

The divergence-free WKB method proposed in our previous papers (T. Hyouguchi, S. Adachi and M. Ueda, Phys. Rev. Lett. 88 (2002), 170404 and T. Hyouguchi, R. Seto, M. Ueda and S. Adachi, Ann. of Phys. 312 (2004), 177) yields a WKB wave function free from divergence for any real argument x ∈ ℝ even around the classical turning point. It is expected in asymptotic analysis that any WKB method for differential equations can be translated in one-to-one manner to the corresponding steepest descent method for integrals. This paper presents newly a steepest descent method that corresponds to our divergence-free WKB method. The experience of this translation lets us notice that the key to improving the traditional steepest descent method for general integrals to become divergence-free is the choice of integration variable to express a given integral.