Abstract:
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.
This research is supported by the Russian Foundation
for Basic Research (Grant No. 18-31-00273) and was also supported by the Federal Target Program (No. AAAA-A17-117021310377-1).
Citation:
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”, TMF, 204:2 (2020), 171–180; Theoret. and Math. Phys., 204:2 (2020), 984–992
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\paper Airy function and transition between the semiclassical and harmonic oscillator approximations for one-dimensional bound states
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\pages 171--180
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\jour Theoret. and Math. Phys.
\yr 2020
\vol 204
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\pages 984--992
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Anna V. Tsvetkova, “Lagrangian Manifolds in the Theory of Wave Beams and Solutions of the Helmholtz Equation”, Regul. Chaotic Dyn., 29:6 (2024), 866–885
V. V. Rykhlov, “Efficient semiclassical approximation for bound states in graphene in magnetic field with a small trigonal warping correction”, Math. Notes, 116:6 (2024), 1339–1349
A. A. Fedotov, “Close Turning Points and the Harper Operator”, Math. Notes, 113:5 (2023), 741–746
A. A. Fedotov, “Complex WKB method (one-dimensional linear problems on the complex plane)”, Math Notes, 114:5-6 (2023), 1418
S. Yu. Dobrokhotov, A. V. Tsvetkova, “Global asymptotics for functions of parabolic cylinder and solutions of the Schrödinger equation with a potential in the form of a nonsmooth double well”, Russ. J. Math. Phys., 30:1 (2023), 46
A. Yu. Anikin, S. Yu. Dobrokhotov, A. A. Shkalikov, “On Expansions in the Exact and Asymptotic Eigenfunctions of the One-Dimensional Schrödinger Operator”, Math. Notes, 112:5 (2022), 623–641
A. A. Fedotov, “Semiclassical Asymptotics for a Difference Schrödinger Equation with Two Coalescent Turning Points”, Math. Notes, 109:6 (2021), 990–994
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