Exact solution of the Schrödinger equation for the inverse square root potential

Abstract

We present the exact solution of the stationary Schrödinger equation for the potential . Each of the two fundamental solutions that compose the general solution of the problem is given by a combination with non-constant coefficients of two confluent hypergeometric functions of a shifted argument. Alternatively, the solution is written through the first derivative of a tri-confluent Heun function. Apart from the quasi-polynomial solutions provided by the energy specification , we discuss the bound-state wave functions vanishing both at infinity and in the origin. The exact spectrum equation involves two Hermite functions of non-integer order which are not polynomials. An accurate approximation for the spectrum providing a relative error less than 10−3 is . Each of the wave functions of bound states in general involves a combination with non-constant coefficients of two confluent hypergeometric and two non-integer order Hermite functions of a scaled and shifted coordinate.