Geodesic
On Expansions in the Exact and Asymptotic Eigenfunctions of the One-Dimensional Schr\"odinger Operator
Matematičeskie zametki, Tome 112 (2022) no. 5, pp. 644-664

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The one-dimensional Schrödinger operator with potential growing at infinity and with a semiclassical small parameter is considered. We obtain estimates via powers of the small parameter for the remainder in the expansion of smooth sufficiently rapidly decaying functions in the exact and asymptotic eigenfunctions. For the asymptotic eigenfunctions, we use a global representation in the form of an Airy function.
Keywords: eigenfunction, asymptotic eigenfunction, Schrödinger operator, semiclassical asymptotics. 
@article{MZM_2022_112_5_a1,
     author = {A. Yu. Anikin and S. Yu. Dobrokhotov and A. A. Shkalikov},
     title = {On {Expansions} in the {Exact} and {Asymptotic} {Eigenfunctions} of the {One-Dimensional} {Schr\"odinger} {Operator}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {644--664},
     publisher = {mathdoc},
     volume = {112},
     number = {5},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2022_112_5_a1/}
}
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A. Yu. Anikin; S. Yu. Dobrokhotov; A. A. Shkalikov. On Expansions in the Exact and Asymptotic Eigenfunctions of the One-Dimensional Schr\"odinger Operator. Matematičeskie zametki, Tome 112 (2022) no. 5, pp. 644-664. http://geodesic.mathdoc.fr/item/MZM_2022_112_5_a1/