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The semiclassical limit of eigenfunctions of the Schrödinger equation and the Bohr-Sommerfeld quantization condition, revisited

Author: D. R. Yafaev
Original publication: Algebra i Analiz, tom 22 (2010), nomer 6.
Journal: St. Petersburg Math. J. 22 (2011), 1051-1067
MSC (2010): Primary 47A40, 81U05
Published electronically: August 22, 2011
MathSciNet review: 2760094
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Abstract | References | Similar Articles | Additional Information

Abstract: 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 asymptotic formulas for eigenfunctions are justified, and the Bohr-Sommerfeld quantization condition is recovered.

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  • 1. V. M. \cyr{B}abich and V. S. \cyr{B}uldyrev, Asimptoticheskie metody v zadachakh difraktsii korotkikh voln. Tom l, Izdat. “Nauka”, Moscow, 1972 (Russian). Metod etalonnykh zadach. [The method of canonical problems]; With the collaboration of M. M. Popov and I. A. Molotkov. MR 0426630
    V. M. Babič and V. S. Buldyrev, Short-wavelength diffraction theory, Springer Series on Wave Phenomena, vol. 4, Springer-Verlag, Berlin, 1991. Asymptotic methods; Translated from the 1972 Russian original by E. F. Kuester. MR 1245488
  • 2. M. V. Fedoryuk, Asimptoticheskie metody dlya lineinykh obyknovennykh differentsialnykh uravnenii, \cyr Spravochnaya Matematicheskaya Biblioteka. [Mathematical Reference Library], “Nauka”, Moscow, 1983 (Russian). MR 732787
  • 3. V. P. Maslov and M. V. Fedoryuk, Kvaziklassicheskoe priblizhenie dlya uravnenii kvantovoi mekhaniki, Izdat. “Nauka”, Moscow, 1976 (Russian). MR 0461590
    V. P. Maslov and M. V. Fedoriuk, Semiclassical approximation in quantum mechanics, Mathematical Physics and Applied Mathematics, vol. 7, D. Reidel Publishing Co., Dordrecht-Boston, Mass., 1981. Translated from the Russian by J. Niederle and J. Tolar; Contemporary Mathematics, 5. MR 634377
  • 4. B. Helffer, A. Martinez, and D. Robert, Ergodicité et limite semi-classique, Comm. Math. Phys. 109 (1987), no. 2, 313–326 (French, with English summary). MR 880418
  • 5. B. Helffer and D. Robert, Puits de potentiel généralisés et asymptotique semi-classique, Ann. Inst. H. Poincaré Phys. Théor. 41 (1984), no. 3, 291–331 (French, with English summary). MR 776281
  • 6. L. D. Landau and E. M. Lifšic, Mekhanika, Theoretical Physics, Vol. I, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1958 (Russian). MR 0102191
    L. D. Landau and E. M. Lifshitz, Course of theoretical physics. Vol. 1, 3rd ed., Pergamon Press, Oxford-New York-Toronto, Ont., 1976. Mechanics; Translated from the Russian by J. B. Skyes and#J. S. Bell. MR 0475051
  • 7. F. W. J. Olver, Asymptotics and special functions, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Computer Science and Applied Mathematics. MR 0435697
  • 8. Barry Simon, Semiclassical analysis of low lying eigenvalues. II. Tunneling, Ann. of Math. (2) 120 (1984), no. 1, 89–118. MR 750717, 10.2307/2007072

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Additional Information

D. R. Yafaev
Affiliation: Irmar, Université de Rennes I, Campus de Beaulieu, 35042 Rennes Cedex, France

Keywords: Schrödinger equation, potential well, Airy functions, Green–Liouville approximation, Bohr–Sommerfeld quantization condition, semiclassical Weyl formula
Received by editor(s): August 5, 2010
Published electronically: August 22, 2011
Additional Notes: Partially supported by the project NONAa, ANR-08-BLANC-0228
Dedicated: To Vasiliĭ Mikhaĭlovich Babich on his 80th birthday
Article copyright: © Copyright 2011 American Mathematical Society