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

We consider the one-dimensional Schrodinger 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.