Exact solution of the Schrödinger equation for the potential $V_0/{\sqrt{x}}$

A. M. Ishkhanyan

Published 2015-08-28Version 1

We present the exact solution of the stationary Schr\"odinger equation for inverse square root potential $V=V_0/{\sqrt{x}}$. 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 standard quasi-polynomial eigensolutions providing the spectrum $E_n=E_1{n^{-2/3}}$, we discuss the case of bound-state wave functions vanishing both at infinity and in the origin. The exact spectrum equation for this case involves two Hermite functions of non-integer order which are not polynomials. An accurate approximation for the spectrum providing absolute error less than $10^{-3}$ is $E_n=E_1{(n-1/(2 \pi))^{-2/3}}$ . 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.