Geodesic
The semiclassical limit of eigenfunctions of the Schrödinger equation and the Bohr–Sommerfeld quantization condition, revisited
Algebra i analiz, Tome 22 (2010) no. 6, pp. 270-291

Voir la notice de l'article provenant de la source Math-Net.Ru

The semiclassical limit, as the Planck constant $\hbar$ tends to 0, of bound states of a quantum particle in a one-dimensional potential well is considered. The semiclassical asymptotics of eigenfunctions is justified, and the Bohr–Sommerfeld quantization condition is recovered.
Keywords: Schrödinger equation, potential well, Airy functions, Green–Lioville approximation, Bohr–Sommerfeld quantization condition, semiclassical Weyl formula. 
D. R. Yafaev. The semiclassical limit of eigenfunctions of the Schrödinger equation and the Bohr–Sommerfeld quantization condition, revisited. Algebra i analiz, Tome 22 (2010) no. 6, pp. 270-291. http://geodesic.mathdoc.fr/item/AA_2010_22_6_a11/
@article{AA_2010_22_6_a11,
     author = {D. R. Yafaev},
     title = {The semiclassical limit of eigenfunctions of the {Schr\"odinger} equation and the {Bohr{\textendash}Sommerfeld} quantization condition, revisited},
     journal = {Algebra i analiz},
     pages = {270--291},
     year = {2010},
     volume = {22},
     number = {6},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/AA_2010_22_6_a11/}
}
TY  - JOUR
AU  - D. R. Yafaev
TI  - The semiclassical limit of eigenfunctions of the Schrödinger equation and the Bohr–Sommerfeld quantization condition, revisited
JO  - Algebra i analiz
PY  - 2010
SP  - 270
EP  - 291
VL  - 22
IS  - 6
UR  - http://geodesic.mathdoc.fr/item/AA_2010_22_6_a11/
LA  - en
ID  - AA_2010_22_6_a11
ER  - 
%0 Journal Article
%A D. R. Yafaev
%T The semiclassical limit of eigenfunctions of the Schrödinger equation and the Bohr–Sommerfeld quantization condition, revisited
%J Algebra i analiz
%D 2010
%P 270-291
%V 22
%N 6
%U http://geodesic.mathdoc.fr/item/AA_2010_22_6_a11/
%G en
%F AA_2010_22_6_a11

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

[2] Fedoryuk M. V., Asimptoticheskie metody dlya lineinykh obyknovennykh differentsialnykh uravnenii, Nauka, M., 1983 | MR | Zbl

[3] Maslov V. P., Fedoryuk M. V., Kvaziklassicheskoe priblizhenie dlya uravnenii kvantovoi mekhaniki, Nauka, M., 1976 | MR

[4] Helffer B., Martinez A., Robert D., “Ergodicité et limite semi-classique”, Comm. Math. Phys., 109 (1987), 313–326 | DOI | MR | Zbl

[5] Helffer B., Robert D., “Puits de potentiel généralisés et asymptotique semi-classique”, Ann. Inst. H. Poincaré Phys. Théor., 41:3 (1984), 291–331 | MR | Zbl

[6] Landau L. D., Lifshits E. M., Teoreticheskaya fizika, v. 1, Mekhanika, Fizmatlit, M., 1958 | Zbl

[7] Olver F. W. J., Asymptotics and special functions, Acad. Press, New York–London, 1974 | MR | Zbl

[8] Simon B., “Semiclassical analysis of low lying eigenvalues. II. Tunneling”, Ann. of Math. (2), 120 (1984), 89–118 | DOI | MR | Zbl