Exact solution of the Schr\"odinger equation for the inverse square root potential $V_0/{\sqrt{x}}$

TL;DR

This paper derives exact solutions for the Schrödinger equation with a $V_0/\sqrt{x}$ potential, expressing solutions via hypergeometric and Heun functions, and provides approximate energy spectra with high accuracy.

Contribution

It presents the first exact analytical solutions for the inverse square root potential in quantum mechanics, including explicit wave functions and spectrum approximations.

Findings

Exact solutions involve confluent hypergeometric and Heun functions.

Bound state energies follow a specific $n^{-2/3}$ dependence.

An accurate spectrum approximation has less than 0.1% error.

Abstract

We present the exact solution of the stationary Schr\"odinger equation equation for the 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 quasi-polynomial solutions provided by the energy specification $E_n=E_1{n^{-2/3}}$, 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 $E_n=E_1{(n-1/(2 \pi))^{-2/3}}$ . 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.