Geodesic
Asymptotic eigenfunctions of the “bouncing ball” type for the two-dimensional Schrödinger operator with a symmetric potential
Teoretičeskaâ i matematičeskaâ fizika, Tome 199 (2019) no. 3, pp. 429-444 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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~{\textquotedblleft}bouncing ball{\textquotedblright} 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},
     year = {2019},
     volume = {199},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2019_199_3_a5/}
}
TY  - JOUR
AU  - A. I. Klevin
TI  - Asymptotic eigenfunctions of the “bouncing ball” type for the two-dimensional Schrödinger operator with a symmetric potential
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2019
SP  - 429
EP  - 444
VL  - 199
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TMF_2019_199_3_a5/
LA  - ru
ID  - TMF_2019_199_3_a5
ER  - 
%0 Journal Article
%A A. I. Klevin
%T Asymptotic eigenfunctions of the “bouncing ball” type for the two-dimensional Schrödinger operator with a symmetric potential
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2019
%P 429-444
%V 199
%N 3
%U http://geodesic.mathdoc.fr/item/TMF_2019_199_3_a5/
%G ru
%F TMF_2019_199_3_a5
A. I. Klevin. Asymptotic eigenfunctions of the “bouncing ball” type for the two-dimensional Schrödinger 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/

[1] V. M. Babich, V. S. Buldyrev, Asimptoticheskie metody v zadachakh difraktsii korotkikh voln, Nauka, M., 1972 | MR | MR

[2] K. Nakamura, T. Harayama, Quantum Chaos and Quantum Dots, Oxford Univ. Press, Oxford, 2004

[3] V. P. Maslov, The Complex WKB Method for Nonlinear Equations I: Linear Theory, Progress in Physics, 16, Birkhäuser, Basel, 1994 | DOI | MR

[4] V. V. Belov, S. Yu. Dobrokhotov, “Kvaziklassicheskie asimptotiki Maslova s kompleksnymi fazami. I. Obschii podkhod”, TMF, 92:2 (1992), 215–254 | DOI | MR

[5] V. V. Belov, O. S. Dobrokhotov, S. Yu. Dobrokhotov, “Izotropnye tory, kompleksnyi rostok i indeks Maslova, normalnye formy i kvazimody mnogomernykh spektralnykh zadach”, Matem. zametki, 69:4 (2001), 483–514 | DOI | DOI | MR | Zbl

[6] J. V. Ralston, “On the construction of quasimodes associated with stable periodic orbits”, Commun. Math. Phys., 51:3 (1976), 219–242 | DOI | MR

[7] V. I. Arnold, V. V. Kozlov, A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, Encyclopaedia of Mathematical Sciences, 3, Springer, Berlin, 2006 | MR

[8] V. G. Bagrov, V. V. Belov, A. Yu. Trifonov, A. A. Yevseyevich, “Quantization of closed orbits in Dirac theory by Maslov's complex germ method”, J. Phys. A: Math. Gen., 27:3 (1994), 1021–1043 | DOI | MR

[9] V. G. Bagrov, V. V. Belov, A. Yu. Trifonov, A. A. Yevseyevich, “Quasi-classical spectral series of the Dirac operators corresponding to quantized two-dimensional Lagrangian tori”, J. Phys. A: Math. Gen., 27:15 (1994), 5273–5306 | DOI | MR

[10] V. V. Belov, V. M. Olive, J. L. Volkova, “The Zeeman effect for the ‘anisotropic hydrogen atoms’ in the complex WKB approximation: II. Quantization of two-dimensional Lagrangian tori (with focal points) for the Pauli operator with spin-orbit interaction”, J. Phys. A: Math. Gen., 28:20 (1995), 5811–5829 | DOI | MR

[11] V. V. Belov, V. M. Olive, J. L. Volkova, “The Zeeman effect for the ‘anistropic hydrogen atom’ in the complex WKB approximation: I. Quantization of closed orbits for the Pauli operator with spin-orbit interaction”, J. Phys. A: Math. Gen., 28:20 (1995), 5799–5810 | DOI | MR

[12] I. M. Gelfand, V. B. Lidskii, “O strukture oblastei ustoichivosti lineinykh kanonicheskikh sistem differentsialnykh uravnenii s periodicheskimi koeffitsientami”, UMN, 10:1(63) (1955), 3–40 | MR | Zbl

[13] V. A. Yakubovich, V. M. Starzhinskii, Lineinye differentsialnye uravneniya s periodicheskimi koeffitsientami i ikh prilozheniya, Nauka, M., 1972 | MR | Zbl

[14] F. W. J. Olver, Asymptotic and Special Functions, A. K. Peters, Wellesley, MA, 1997 | MR

[15] C. Chester, B. Friedman, F. Ursell, “An extension of the method of steepest descent”, Math. Proc. Cambridge Phil. Soc., 53:3 (1957), 599–611 | DOI | MR

[16] S. Yu. Slavyanov, Asimptotika reshenii odnomernogo uravneniya Shredingera, Izd-vo Leningr. un-ta, L., 1991 | MR

[17] C. Yu. Dobrokhotov, D. S. Minenkov, S. B. Shlosman, “Asimptotika volnovykh funktsii statsionarnogo uravneniya Shredingera v kamere Veilya”, TMF, 197:2 (2018), 269–278 | DOI | DOI

[18] C. Yu. Dobrokhotov, A. V. Tsvetkova, “O lagranzhevykh mnogoobraziyakh svyazannykh s asimptotikoi polinomov Ermita”, Matem. zametki, 104:6 (2018), 835–850 | DOI | DOI