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Airy functions and transition between semiclassical and harmonic oscillator approximations for one-dimensional bound states. (English. Russian original) Zbl 1453.81024

Theor. Math. Phys. 204, No. 2, 984-992 (2020); translation from Teor. Mat. Fiz. 204, No. 2, 171-180 (2020).
Summary: 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.

MSC:

81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
35P05 General topics in linear spectral theory for PDEs
35B40 Asymptotic behavior of solutions to PDEs
33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\)
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References:

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