WKB spectrum of a square-root half-line well ($E_n\propto (n+3/4{)}^{2/5}$)

For the half-line square-root potential well, the dimensionless action integral isHence $E_n/E_{\ast }=[15\pi (n+3/4)/8{]}^{2/5}$. The ground-state estimate is $1.811694E_{\ast }$. This is a semiclassical estimate, not a variational upper bound.