Abstract
The motion of a particle in the field of an electromagnetic monopole (in the Coulomb–Dirac field) perturbed by an axially symmetric potential after quantum averaging is described by an integrable system. Its Hamiltonian can be written in terms of the generators of an algebra with quadratic commutation relations. We construct the irreducible representations of this algebra in terms of second-order differential operators; we also construct its hypergeometric coherent states. We use these states in the first-order approximation with respect to the perturbing field to obtain the integral representation of the eigenfunctions of the original problem in terms of solutions of the model Heun-type second-order ordinary differential equation and present the asymptotic approximation of the corresponding eigenvalues.
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REFERENCES
V. P. Maslov, Operator Methods [in Russian], Nauka, Moscow (1973); English transl.: Operational Methods, Mir, Moscow (1976).
M. V. Karasev, Funct.Anal.Appl., 18, 140–142 (1984).
M. V. Karasev and V. P. Maslov, Russ.Math.Surveys, 39, No. 6, 133–205 (1984); M. V. Karasev, “Quantum reduction to orbits of symmetry algebra and the Ehrenfest problem [in Russian],” Preprint ITF-87-157P, Inst. Theor. Phys., Ukrainian Acad. Sci., Kiev (1987).
J. R. Klauder and B. S. Skagerstam, Coherent States: Applications in Physics and Mathematics, World Scientific, Singapore (1985); A. M. Perelomov, Generalized Coherent States and Their Applications, Springer, Berlin (1986).
M. Karasev, Russ.J.Math.Phys., 3, 393–400 (1995).
M. V. Karasev, Lett.Math.Phys., 56, 229–269 (2001).
M. V. Karasev and E. M. Novikova, Russ.Math.Surveys, 49, No. 5, 179–180 (1994).
M. V. Karasev and E. M. Novikova, Theor.Math.Phys., 108, 1119–1159 (1996).
M. V. Karasev and E. M. Novikova, J.Math.Sci. (New York), 95, 2703–2798 (1999).
M. V. Karasev and E. M. Novikova, “Adapted connections, Hamilton dynamics, geometric phases, and quantization over isotopic submanifolds,” in: Coherent Transform,Quantization,and Poisson Geometry (M.V. Karasev, ed.), Amer. Math. Soc., Providence, R. I. (1998), pp. 1–202.
M. V. Karasev and E. M. Novikova, Math.Notes, 70, 779–797 (2001).
M. V. Karasev and E. M. Novikova, Math.Notes, 72, 48–65 (2002).
V. P. Maslov, Theor.Math.Phys., 33, 960–976 (1977); M. V. Karasev and V. P. Maslov, Russ.Math.Surveys, 45, No. 5, 51–98 (1990); Nonlinear Poisson Brackets: Geometry and Quantization [in Russian], Nauka, Moscow (1991); English transl., Amer. Math. Soc., Providence, R. I. (1993).
E. K. Sklyanin, Funct.Anal.Appl., 16, 263–270 (1983); 17, 273–284 (1983).
H. V. McIntosh and A. Cisneros, J.Math.Phys., 11, 896–916 (1970).
I. Mladenov and V. Tsanov, J.Phys.A, 20, 5865–5871 (1987).
T. Iwai and Y. Uwano, J.Phys.A, 21, 4083–4104 (1988).
I. Mladenov, Ann.Inst.H.Poincaré, 50, 219–227 (1989).
A. Yoshioka and K. Ii, J.Math.Phys., 31, 1388–1394 (1990).
H. Bateman and A. Erdélyi, eds., Higher Transcendental Fun tions (Based on notes left by H. Bateman), Vol. 1, McGraw-Hill, New York (1953).
H. Bateman and A. Erdélyi, eds., Higher Transcendental Fun tions (Based on notes left by H. Bateman), Vol. 2, McGraw-Hill, New York (1953).
E. Kamke, Differentialgleichungen, Lösunsmethoden und Lösungen, Acad. Verlag, Leipzig (1959).
P. Kustaanheimo, Ann.Univ.Turku A, 73, No. 1, 3–7 (1964); P. Kustaanheimo and E. Stiefel, J.Reine Angew. Math., 218, 204–219 (1965).
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Karasev, M.V., Novikova, E.M. Algebra with Quadratic Commutation Relations for an Axially Perturbed Coulomb–Dirac Field. Theoretical and Mathematical Physics 141, 1698–1724 (2004). https://doi.org/10.1023/B:TAMP.0000049763.86662.16
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DOI: https://doi.org/10.1023/B:TAMP.0000049763.86662.16