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We present the exact solution of the stationary Schr\"odinger equationequation for the potential $V=V_0/{\sqrt{x}}$. Each of the two fundamentalsolutions that compose the general solution of the problem is given by acombination with non-constant coefficients of two confluent hypergeometricfunctions of a shifted argument. Alternatively, the solution is written throughthe first derivative of a tri-confluent Heun function. Apart from thequasi-polynomial solutions provided by the energy specification$E_n=E_1{n^{-2/3}}$, we discuss the bound-state wave functions vanishing bothat infinity and in the origin. The exact spectrum equation involves two Hermitefunctions of non-integer order which are not polynomials. An accurateapproximation 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 statesin general involves a combination with non-constant coefficients of twoconfluent hypergeometric and two non-integer order Hermite functions of ascaled and shifted coordinate.

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