Geodesic
Representation of exact and semiclassical eigenfunctions via coherent states. Hydrogen atom in a magnetic field
Teoretičeskaâ i matematičeskaâ fizika, Tome 108 (1996) no. 3, pp. 339-387 Cet article a éte moissonné depuis la source Math-Net.Ru

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A coherent states technique is applied to obtain global formulas for eigenfunctions as well as for solutions of the Cauchy problem, including a path integral representation. The reduction of coherent states by symmetry groups is studied using a transformation from “Bessel” states to “hypergeometric” ones. The eigenfunctions of the Hamiltonian of the hydrogen atom in a homogeneous magnetic field are represented via Bessel coherent states. In the case of small field, after a quantum averaging, the Hamiltonian is expressed by means of generators with quadratic commutation relations. Irreducible representations of this quadratic algebra are realized on the hypergeometric states. The notion of deformed hypergeometric states is also introduced for this quadratic algebra; it is an analog of squeezed Gaussian packets usually related to the Heisenberg algebra. The asymptotics of eigenfunctions with respect to small field and to high leading quantum number is derived using these states and their deaveraging. Explicite formulas for the Zeeman spectrum splitting are obtained up to the fourth order with respect to the field, for lower and upper levels in the claster as well, including the case of “passing through the center”. 
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M. V. Karasev; E. M. Novikova. Representation of exact and semiclassical eigenfunctions via coherent states. Hydrogen atom in a magnetic field. Teoretičeskaâ i matematičeskaâ fizika, Tome 108 (1996) no. 3, pp. 339-387. http://geodesic.mathdoc.fr/item/TMF_1996_108_3_a0/

[1] H. Friedrich, D. Wintgen, Phys. Reports, Rev. Section of Phys. Letters, 189, no. 2, North Holland, Amsterdam, 1989, 37 | MR

[2] J. Avron, Ann. Phys., 131 (1981), 73 | DOI | MR

[3] J. Avron, I. Herbst, B. Simon, Commun. Math. Phys., 79 (1981), 529 | DOI | MR | Zbl

[4] M. Kummer, A. W. Saenz, Commun. Math. Phys., 162:3 (1994), 447 | DOI | MR | Zbl

[5] V. I. Arnold, Dopolnitelnye glavy teorii obyknovennykh differentsialnykh uravnenii, Nauka, M., 1978 | MR

[6] E. A. Solovev, Pisma v ZhETF, 34 (1981), 278

[7] E. A. Solovev, ZhETF, 82:6 (1982), 1762

[8] D. R. Herrick, Phys. Rev. A, 26:1 (1982), 323 | DOI | MR

[9] D. Delande, J. C. Gay, J. Phys. B: At. Mol. Phys., 17 (1984), L 335 | DOI | MR

[10] T. P. Grozdanov, H. S. Taylor, J. Phys. B: At. Mol. Phys., 19:24 (1986), 4075 | DOI | MR

[11] H. Hasegava, M. Robnik, G. Wunner, Progr. Theor. Phys. Suppl., 1989, no. 98, 198 | DOI

[12] M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics, Springer-Verlag, 1990 | MR

[13] E. G. Kalnins, W. Miller, P. Winternitz, SIAM J. Appl. Math., 30:4 (1976), 630 | DOI | MR | Zbl

[14] A. Weinstein, Duke Math. J., 44 (1977), 883 | DOI | MR | Zbl

[15] M. V. Karasev, Funkts. analiz i ego prilozh., 18:2 (1984), 65 | MR | Zbl

[16] M. V. Karasev, V. P. Maslov, UMN, 39:6 (1984), 115 | MR | Zbl

[17] E. K. Sklyanin, Funkts. analiz i ego prilozh., 16:4 (1982), 27 | MR | Zbl

[18] M. V. Karasev, E. M. Novikova, UMN, 49:5 (1994), 169 | MR | Zbl

[19] M. V. Karasev, Symplectic Geom. and Math. Phys., Actes du colloque en l'honneur de J.-M.Souriau, eds. P. Donato et al., Birkhäuser, Boston, 1991, 234 | DOI | MR

[20] E. J. Heller, J. Chem. Phys., 62 (1975), 1544 | DOI

[21] A. Voros, Phys. Rev. A, 40:12 (1989), 6814 | DOI | MR

[22] J. Kurchan, P. Leboeuf, M. Saraceno, Phys. Rev. A, 40 (1989), 6800 | DOI | MR

[23] J. Sov. Math., 59:5 (1992), 1053 | DOI | MR

[24] M. V. Karasev, To the Maslov theory of semiclassical asymptotics. Examples of a new global quantization formula applications, Preprint Inst. Theor. Phys. Kiev. ITP-89-78 E, 1989

[25] M. V. Karasev, Quantum Field Theory, Quantum Mechanics and Quantum Optics, Part I, Nova Sci. Publ., New York, 1991, 189

[26] M. V. Karasev, Contemp. Math., 179 (1994), 83 | DOI | MR | Zbl

[27] M. V. Karasev, Funkts. analiz i ego prilozh., 24:2 (1990), 104 | MR | Zbl

[28] M. Karasev, M. Kozlov, J. Math. Phys., 34:11 (1993), 4986 | DOI | MR | Zbl

[29] M. V. Karasev, M. B. Kozlov, Funkts. analiz i ego prilozh., 28:4 (1994), 16 | MR | Zbl

[30] V. A. Fok, Izv. AN SSSR. Otd. mat. est. nauk, 1935, no. 2, 169

[31] J. Moser, Commun. Pure Appl. Math., 23 (1970), 609 | DOI | MR | Zbl

[32] J.-M. Souriau, “Sur la varieté de Kepler”, Symposia Math., XIV, Academic Press, 1974 | MR | Zbl

[33] P. Kustaanheimo, E. Stiefel, J. Reine und Angew. Math., 218 (1965), 204 | MR | Zbl

[34] Coherent Transforms, Quantization and Poisson Geometry, Adv. Modern Math., ed. M. V. Karasev, AMS, N.Y., 1997 (to appear) | MR

[35] M. V. Karasev, TMF, 61:1 (1984), 118 | MR | Zbl

[36] M. V. Karasev, Russ. J. Math. Phys., 1:4 (1993), 523 | MR | Zbl

[37] M. V. Karasev, Russ. J. Math. Phys., 3:3 (1995), 393 | MR | Zbl

[38] V. I. Arnold, Funkts. analiz i ego prilozh., 6:2 (1972), 12 | MR | Zbl

[39] V. V. Belov, J. L. Volkova, Russ. J. Math. Phys., 1:4 (1993), 409 | MR | Zbl

[40] V. V. Belov, V. M. Olivé, J. L. Volkova, J. Phys. A, 28 (1995), 5799 | DOI | MR | Zbl

[41] P. N. Zhevandrov, J. Math. Phys., 35:4 (1994), 1597 | DOI | MR | Zbl

[42] M. V. Karasev, Matem. zametki, 60:6 (1996) | DOI | Zbl

[43] M. V. Karasev, Quantiz., Coherent States and Complex Struct., eds. J.-P. Antoine et al., Plenum, N.Y., 1996, 185 | MR

[44] M. V. Karasev, E. M. Novikova, Quantiz., Coherent States and Complex Struct., eds. J.-P. Antoine et al., Plenum, N.Y., 1996, 201 | MR

[45] M. V. Karasev, E. M. Novikova, UMN, 50:4 (1995), 100 | MR

[46] A. I. Baz, Ya. B. Zeldovich, A. M. Perelomov, Rasseyanie, reaktsii i raspady v nerelyativistskoi kvantovoi mekhanike, Nauka, M., 1971 | Zbl

[47] Ya. I. Granovskii, A. S. Zhedanov, I. M. Lutzenko, J. Phys., A 24 (1991), 3887 | MR

[48] J. H. Rawnsley, Quart. J. Math. Oxford, 28 (1977), 403 | DOI | MR | Zbl

[49] M. Horowski, A. Odzijewicz, Geometry of the Kepler system in coherent states approach, Preprint Inst. of Physics. Warsaw Univ. Division Bialystok, 1993 | MR | Zbl

[50] M. V. Karasev, Funkts. analiz i ego prilozh., 20:1 (1986), 21 | MR | Zbl

[51] M. V. Karasev, V. P. Maslov, Nelineinye skobki Puassona. Geometriya i kvantovanie, Nauka, M., 1991 | MR | Zbl

[52] E. Kamke, Spravochnik po obyknovennym differentsialnym uravneniyam, Nauka, M., 1971 | MR

[53] V. F. Lazutkin, Itogi nauki i tekhn. Sovr. probl. matem. Fund. napravl., 34, VINITI, M., 1988, 135 | MR | Zbl

[54] G. Beitmen, A. Erdeii, Vysshie transtsendentnye funktsii, T. I, Nauka, M., 1973 ; Т. II, 1974 | MR

[55] J. R. Klauder, J. Math. Phys., 4 (1963), 1058 | DOI | MR | Zbl

[56] F. A. Berezin, Izv. AN SSSR. Ser. matem., 36 (1972), 1134 | MR | Zbl

[57] A. M. Perelomov, Obobschennye kogerentnye sostoyaniya i ikh primeneniya, Nauka, M., 1987 | MR

[58] J. Rawnsley, M. Cahen, S. Gutt, J. Geom. Phys., 7 (1990), 45 | DOI | MR | Zbl

[59] A. Odzijewicz, Commun. Math. Phys., 114 (1988), 577 | DOI | MR | Zbl

[60] S. T. Ali, J.-P. Antoine, Quantization and Infinite-Dimen. Systems, Plenum, N.Y., 1994, 133 | DOI | MR | Zbl

[61] M. V. Karasev, Russ. J. Math. Phys., 2:4 (1994), 445 | MR | Zbl

[62] F. A. Berezin, TMF, 6:2 (1971), 194 | MR | Zbl

[63] J. R. Klauder, Phys. Rev. Lett., 56 (1986), 897 | DOI | MR

[64] A. C. Cadavid, M. Nakashima, Lett. Math. Phys., 23 (1991), 111 | DOI | MR | Zbl

[65] F. A. Berezin, M. S. Marinov, Ann. Phys., 104 (1977), 336 | DOI | Zbl