Geodesic
Asymptotic eigenfunctions of the~``bouncing ball'' type for the~two-dimensional Schr\"odinger operator with a~symmetric potential
Teoretičeskaâ i matematičeskaâ fizika, Tome 199 (2019) no. 3, pp. 429-444

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We construct asymptotic eigenfunctions for the two-dimensional Schrödinger operator with a potential in the form of a well that is mirror-symmetric with respect to a line. These functions correspond to librations on this line between two focal points. According to the Maslov complex germ theory, the asymptotic eigenfunctions in the direction transverse to the line with respect to which the well is symmetric have the form of the appropriate Hermite–Gauss mode. We obtain a global Airy-function representation for the asymptotic eigenfunctions in the longitudinal direction.
Keywords: stationary Schrödinger equation, spectrum, asymptotics, canonical Maslov operator, complex germ, Airy function. 
@article{TMF_2019_199_3_a5,
     author = {A. I. Klevin},
     title = {Asymptotic eigenfunctions of the~``bouncing ball'' type for the~two-dimensional {Schr\"odinger} operator with a~symmetric potential},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {429--444},
     publisher = {mathdoc},
     volume = {199},
     number = {3},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2019_199_3_a5/}
}
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A. I. Klevin. Asymptotic eigenfunctions of the~``bouncing ball'' type for the~two-dimensional Schr\"odinger operator with a~symmetric potential. Teoretičeskaâ i matematičeskaâ fizika, Tome 199 (2019) no. 3, pp. 429-444. http://geodesic.mathdoc.fr/item/TMF_2019_199_3_a5/