Geodesic
Airy function and transition between the semiclassical and harmonic oscillator approximations for one-dimensional bound states
Teoretičeskaâ i matematičeskaâ fizika, Tome 204 (2020) no. 2, pp. 171-180

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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.
Keywords: bound state, Schrödinger operator, semiclassical approximation, asymptotics, eigenfunction, harmonic oscillator, Airy function. 
@article{TMF_2020_204_2_a1,
     author = {A. Yu. Anikin and S. Yu. Dobrokhotov and A. V. Tsvetkova},
     title = {Airy function and transition between the semiclassical and harmonic oscillator approximations for one-dimensional bound states},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {171--180},
     publisher = {mathdoc},
     volume = {204},
     number = {2},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2020_204_2_a1/}
}
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A. Yu. Anikin; S. Yu. Dobrokhotov; A. V. Tsvetkova. Airy function and transition between the semiclassical and harmonic oscillator approximations for one-dimensional bound states. Teoretičeskaâ i matematičeskaâ fizika, Tome 204 (2020) no. 2, pp. 171-180. http://geodesic.mathdoc.fr/item/TMF_2020_204_2_a1/