Airy function and transition between the semiclassical and harmonic oscillator approximations for one-dimensional bound states

We consider the one-dimensional Schrödinger operator with a semiclassical small parameter $h$. We show that the "global" asymptotic form of its bound states in terms of the Airy function "works" not only for excited states $n\sim1/h$ but also for semi-excited states $n\sim1/h^\alpha$, $\alpha>0$, and, moreover, $n$ starts at $n=2$ or even $n=1$ in examples. We also prove that the closeness of such an asymptotic form to the eigenfunction of the harmonic oscillator approximation.