Quantum Painlev´e II solution with approximatedanalytic solution in form of nearly Yukawa potential

Abstract

In this article it has been shown that one dimensional non-stationary Schr¨odinger equation with a specific choice of potential reduces to the quantum Painlev´e II equation and the solution of its riccati form appears
as a dominant term of that potential. Further, we show that Painlev´e II Riccati solution is an equivalent representation of centrifugal expression of radial Schr¨odinger potential. This expression is used to derive
the approximated to the Yukawa potential of radial Schr¨odinger equation which can be solved by applying the Nikiforov-Uvarov method. Finally, we express the approximated form of Yukawa potential explicitly in
terms of quantum Painlev´e II solution.