Quantum Painlev\'e II solution and Approximated analytic solution of the Yukawa Potential

Abstract

We show that one dimensional non-stationary Schr\"odi-nger 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 form of 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 qunatume Painlev\'e II solution.