Airy functions and uniform transition from the oscillator to semiclassical approximation for one-dimensional bound states

We consider a one-dimensional Schrödinger operator with a semiclassical small parameter $h$ with a smooth potential of the form of a potential well. We construct uniform global asymptotics of its eigenfunctions in terms of Airy functions of complex argument. We show that such asymptotics work not only for excited states with numbers $n \sim 1/h$, but also for weakly excited states with numbers $n \sim 1/h^\alpha, 1> \alpha > 0$, and in the examples the corresponding numbers $n$ start with $n = 2$ or even with $n = 1$. We prove the proximity of such asymptotics to the eigenfunction, obtained with the help of the harmonic oscillator.

^* Conference identificator: 931 2278 9257 Password: 045927