Geodesic
Local symmetry algebra of Shrödinger equation for Hydrogen atom
Teoretičeskaâ i matematičeskaâ fizika, Tome 106 (1996) no. 2, pp. 273-284 Cet article a éte moissonné depuis la source Math-Net.Ru

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The complete description of local symmetries (which are differential operators of arbitrary finite order) is given for stationary Shrödinger equation for Hydrogen atom. This is done using the reduction of Shrödinger equation for isotropic harmonic oscillator to one for the Hydrogen atom, which induces the correspondent symmetry algebras reduction. It is shown that all nontrivial local symmetry operators for $n$-dimensional isotropic harmonic oscillator belong to enveloping algebra $U(su(n,C))$ of algebra $su(n,C)$. For Hydrogen atom all nontrivial local symmetries constitute enveloping algebra $U(so(4,C))$ of algebra $so(4,C)$. Basis of $so(4,C)$ consists of rotation group generators and Runge–Lenz-operators. 
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     author = {A. A. Drokin and A. V. Shapovalov and I. V. Shirokov},
     title = {Local symmetry algebra of {Shr\"odinger} equation for {Hydrogen} atom},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {273--284},
     year = {1996},
     volume = {106},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1996_106_2_a7/}
}
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A. A. Drokin; A. V. Shapovalov; I. V. Shirokov. Local symmetry algebra of Shrödinger equation for Hydrogen atom. Teoretičeskaâ i matematičeskaâ fizika, Tome 106 (1996) no. 2, pp. 273-284. http://geodesic.mathdoc.fr/item/TMF_1996_106_2_a7/

[1] Ovsyannikov L. V., Grupovoi analiz differentsialnykh uravnenii, Nauka, M., 1978 | MR

[2] Ibragimov N. Kh., Gruppy preobrazovanii v matematicheskoi fizike, Nauka, M., 1983 | MR

[3] Olver P., Prilozheniya grupp Li k differentsialnym uravneniyam, Mir, M., 1989 | MR | Zbl

[4] Fuschich V. I., Nikitin A. G., Simmetriya uravnenii kvantovoi mekhaniki, Nauka, M., 1990 | MR

[5] Shapovalov V. N., Diff. uravneniya, 16:10 (1980), 1864–1874 | MR | Zbl

[6] Miller U., Simmetriya i razdelenie peremennykh, Mir, M., 1981 | MR

[7] Shapovalov A. V., Shirokov I. V., Izv. vuzov. Fizika, 1991, no. 4, 95–100 | MR

[8] Nikitin A. G., Prilipko A. I., Simmetriya i resheniya uravnenii matematicheskoi fiziki, In-t matematiki Ukrainy, Kiev, 1989, 40–44

[9] Meshkov A. G., TMF, 55:2 (1983), 197–204 | MR | Zbl

[10] Shapovalov A. V., Shirokov I. V., TMF, 92:1 (1992), 3–12 | MR | Zbl

[11] Shapovalov V. N., Izv. vuzov. Fizika, 1977, no. 6, 57–70

[12] Fok V. A., Izv AN SSSR (otd. matem. i estestv. nauk), 1935, no. 2, 169–188 | MR

[13] Bargman V., Z. Physik, 99 (1936), 576 | DOI

[14] Barut A. O., Bohm A., Phys. Rev. B, 15 (1965), 1107–1112 | MR

[15] Dothan Y., Gell-Mann M., Ne'eman Y., Phys. Lett., 15 (1965), 148–151 | DOI | MR

[16] Mukunda N., O'Raifeartaigh L., Sudarshan E., Phys. Rev. Lett., 15 (1965), 1041–1044 | DOI | MR

[17] Malkin I. A., Manko V. I., Pisma v ZhETF, 2 (1965), 230–234 | MR

[18] Malkin I. A., Manko V. I., YaF, 3:2 (1966), 372–382 | MR

[19] Manko V. I., “Simplekticheskaya gruppa i invarianty kvantovykh sistem”, Itogi nauki i tekhniki. Matem. analiz, 22, VINITI, M., 1984, 59–100

[20] Dodonov V. V., Manko V. I., Tr. FIAN SSSR im. P. N. Lebedeva, 183, 1987, 71–181 | MR

[21] Malkin I. A., Manko V. I., Dinamicheskie simmetrii i kogerentnye sostoyaniya kvantovykh sistem, Nauka, M., 1979 | MR

[22] Barut A., Ronchka R., Teoriya predstavlenii grupp i ee prilozheniya, T. 1, 2, Mir, M., 1980 | MR | Zbl

[23] Khart N., Geometricheskoe kvantovanie v deistvii, Mir, M., 1985 | MR

[24] Brychkov Yu. A., Prudnikov A. P., Integralnye preobrazovaniya obobschennykh funktsii, Nauka, M., 1977 | MR | Zbl