Abstract
The vector Sturm-Liouville problem for the system with a small parameter at the derivatives is considered. Asymptotics of eigenvalues and eigenfunctions are constructed for the case in which the symbol of the problem has eigenvalues of variable multiplicity.
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References
L. D. Landau and E. M. Lifshitz, Quantum Mechanics, Vol. III (Fizmatlit, Moscow, 2002).
J. E. Avron and A. Gordon, “Born-Oppenheimer Wave Function near Level Crossing,” Phys. Rev. A. 62, 062504-1-062504-9 (2000).
V. V. Belov, S. Yu. Dobrokhotov, and T. Ya. Tudorovskii, “Quantum and Classical Dynamics of an Electron in Thin Curved Tubes with Spin and External Electromagnetic Fields Taken into Account,” Russ. J. Math. Phys. 11(1), 109–119 (2004).
V. V. Belov, S. Yu. Dobrokhotov, and T. Ya. Tudorovskii, “Asymptotic Solutions of Nonrelativistic Equations of Quantum Mechanics in Curved Nanotubes. I. Reduction to Spatially One-Dimensional Equations,” Teoret. Mat. Fiz. 141(2), 267–303 (2004) [Theoret. and Math. Phys. 141 (2), 1562–1592 (2004)].
M. Reed and B. Simon, Methods of Modern Mathematical Physics, II (Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975).
K. O. Friedrichs, Perturbation of Spectra in Hilbert Space (American Mathematical Society, Providence, 1965).
V. P. Maslov, Perturbation Theory and Asymptotic Methods (Izdat. Moskov. Cos. Univ., Moscow, 1965; Dunod, Paris, 1972).
V. P. Maslov and M. V. Fedoryuk, Semi-Classical Approximations in Quantum Mechanics (Nauka, Moscow, 1976; Reidel, Dordrecht, 1981).
E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations (McGraw-Hill, New York, 1955).
V. V. Kucherenko, “Asymptotic Behavior of the Solution of the System A(x, −ih∂/∂x)u = 0 as h → 0 in the Case of Characteristics with Variable Multiplicity,” Izv. Akad. Nauk SSSR Ser. Mat. 38, 625–662 (1974) [Math. USSR Izv. 8 (3), 631–666 (1974)].
Y. C. De Verdière and B. Parisse, “Singular Bohr-Sommerfeld Rules,” Comm. Math. Phys. 205, 459–500 (1999).
A. Erdélyi, Asymptotic Expansions (Dover, New York, 1956).
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Kryvko, A., Kucherenko, V.V. Semiclassical asymptotics of the vector Sturm-Liouville problem. Russ. J. Math. Phys. 13, 188–202 (2006). https://doi.org/10.1134/S1061920806020051
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DOI: https://doi.org/10.1134/S1061920806020051