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International conference in memory of M. K. Polivanov "Polivanov–90"
December 17, 2020 10:00, Moscow, online

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

S. Yu. Dobrokhotov

Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, Moscow
 Video records: MP4 173.7 Mb

Abstract: 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.