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

We present the exact solution of the stationary Schr"odinger equation equation for the potential $V=V_0\mathrm{/}\sqrt{x}$. 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 $E_n=E_1{n}^{-2\mathrm{/}3}$, 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 $E_n=E_1(n-1\mathrm{/}(2\pi ){)}^{-2\mathrm{/}3}$ . 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.