Abstract
The joint spectrum of the Schrödinger operator of an anisotropic Kepler problem (the component along one of the axes of the diagonal mass tensor differs from the components along the other two axes) and the angular momentum operator are considered. Using the theory of Maslov’s complex germ, series of corresponding semiclassical stationary states (with complex phases) localized in the vicinity of flat disks are constructed. The wave functions of these states, in the direction normal to the plane of the disk, have the form of Hermite–Gaussian functions; in the direction of the polar angle coordinate in the plane of the disk, they oscillate harmonically; in the radial direction in the plane of the disk, their behavior is described by the Airy function of a composed argument: inside the disk, the wave functions oscillate and, outside it, decay. The existence of such states is due to the existence of Floquet solutions of certain differential equations with periodic coefficients.
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Notes
The notation \(+0\) is used for convenience and means an appropriate passage to the limit in formulas. Instead, we could take a sufficiently small positive number.
References
Gutzwiller and Martin C., Chaos in Classical and Quantum Mechanics, Springer-Verlag, New York, 1990.
V. V. Belov, S. Yu. Dobrokhotov, V. Martines Olive, “Some quasiclassical spectral series in a quantum anisotropic Kepler problem”, Dokl. Math., 38:7 (1993), 263–266.
Maslov V. P., The Complex WKB Method for Nonlinear Equations I, Birkhauser, Basel, 1994.
V. V. Belov, S. Yu. Dobrokhotov, “Semiclassical Maslov asymptotics with complex phases. I. General approach”, Theoret. and Math. Phys., 92:2 (1992), 843–868.
V. V. Belov, S. Yu. Dobrokhotov, “The Maslov canonical operator on isotropic manifolds with a complex germ, and its applications to spectral problems”, Dokl. Math., 37:1 (1988), 180–185.
A. Yu. Anikin, S. Yu. Dobrokhotov, V. E. Nazaikinskii, A. V. Tsvetkova, “Uniform asymptotic solution in the form of an Airy function for semiclassical bound states in one-dimensional and radially symmetric problems”, Theoret. and Math. Phys., 201:3 (2019), 382–414.
Dobrokhotov, S. Yu., Nazaikinskii, V. E., “Efficient Formulas for the Maslov Canonical Operator near a Simple Caustic”, Russ. J. Math. Phys., 25:4 (2018), 545–552.
A. I. Klevin, “Asymptotic eigenfunctions of the “bouncing ball” type for the two-dimensional Schrödinger operator with a symmetric potential”, Theoret. and Math. Phys., 199:3 (2019), 849–863.
V. P. Maslov, M. V. Fedoryuk, Semi-Classical Approximation in Quantum Mechanics, D. Reidel Publishing Company, Dordrecht–Boston–London, 1976.
V. P. Maslov, Operator Methods, Nauka, Moscow, 1973.
V. E. Nazaikinskii, B. Yu. Sternin, V. E. Shatalov, Methods of Noncommutative Analysis, Tekhnosfera, Moscow, 2002.
S. Yu. Dobrokhotov, A. I. Shafarevich, “Semiclassical quantization of invariant isotropic manifolds of Hamiltonian systems”, Topological Methods in the Theory of Hamiltonian Systems, Faktorial, Moscow, 1998, 41–114.
S. Yu. Dobrokhotov, V. E. Nazaikinskii, A. I. Shafarevich, “New integral representations of the Maslov canonical operator in singular charts”, Izv. Math., 81:2 (2017), 286–328.
Acknowledgments
The author is grateful to S.Yu. Dobrokhotov for posing the problem and useful discussions.
Funding
The reported study was funded by RFBR, project no. 20-31-90111.
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Klevin, A.I. Uniform Asymptotics in the Form of Airy Functions for Bound States of the Quantum Anisotropic Kepler Problem Localized in a Neighborhood of Annuli. Russ. J. Math. Phys. 29, 47–56 (2022). https://doi.org/10.1134/S1061920822010058
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DOI: https://doi.org/10.1134/S1061920822010058