Asymptotics of the spectrum of the hydrogen atom in a magnetic field near the lower boundaries of spectral clusters

A. V. Pereskokov

Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 73 (2012) no. 2, pp. 277-325.

Bibliographie

[1] Friedrich H., Wintgen D., “The hydrogen atom in a uniform magnetic field — an example of chaos”, Phys. rep., 183:2 (1989), 39–79 | DOI | MR

[2] Lisitsa V. S., “Novoe v effektakh Shtarka i Zeemana dlya atoma vodoroda”, UFN, 153:3 (1987), 379–421 | DOI | MR

[3] Avron J. E., Herbst I. W., Simon B., “Schrödinger operators with magnetic fields. III: Atoms in homogeneous magnetic field”, Comm. Math. Phys., 79:4 (1981), 529–572 | DOI | MR | Zbl

[4] Solovev E. A., “Atom vodoroda v slabom magnitnom pole”, ZhETF, 82:6 (1982), 1762–1771

[5] Herrick D. R., “Symmetry of the quadratic Zeeman effect for hydrogen”, Phys. Rev. A, 26:1 (1982), 323–329 | DOI | MR

[6] Delande D., Gay J. C., “Group theory applied to the hydrogen atom in a strong magnetic field. Derivation of the effective diamagnetic Hamiltonian”, J. Phys. B, 17:11 (1984), L335–L340 | DOI | MR

[7] Belov V. V., Volkova J. L., Investigation of the Zeeman effect in quasiclassical trajectory-coherent approximation, Preprint No 35, Tomsk Scientific Centre, Siberian Division, AS USSR, 1991 | MR

[8] Belov V. V., Olivé V. M., Volkova J. L., “The Zeeman effect for the ‘anistropic hydrogen atom’ in the complex WKB approximation. I: Quantization of closed orbits for the Pauli operator with spin-orbit interaction”, J. Phys. A, 28:20 (1995), 5799–5810 | DOI | MR | Zbl

[9] Karasev M. V., Novikova E. M., “Predstavlenie tochnykh i kvaziklassicheskikh sobstvennykh funktsii cherez kogerentnye sostoyaniya. Atom vodoroda v magnitnom pole”, TMF, 108:3 (1996), 339–387 | DOI | MR | Zbl

[10] Karasev M. V., “Birkhoff resonances and quantum ray method”, Days of Diffraction-2004, Proc. Intern. Seminar, St. Petersburg and Steklov Math. Institute, St. Petersburg, 2004, 114–126 | MR

[11] Karasev M. V., “Noncommutative algebras, nano-structures, and quantum dynamics generated by resonances, I”, Quantum Algebras and Poisson Geometry in Mathematical Physics, Amer. Math. Soc. Trans. Ser. 2, 216, ed. Karasev M., Amer. Math. Soc., Providence, RI, 2005, 1–18 ; II, Adv. Stud. Contemp. Math., 11:1 (2005), 33–56 ; III, Russ. J. Math. Phys., 13:2 (2006), 131–150 | MR | MR | Zbl | DOI | MR | Zbl

[12] Karasev M. V., Novikova E. M., “Algebra s kvadratichnymi kommutatsionnymi sootnosheniyami dlya aksialno-vozmuschennogo polya Kulona–Diraka”, TMF, 141:3 (2004), 424–454 | DOI | MR | Zbl

[13] Karasev M. V., Novikova E. M., “Algebra s polinomialnymi kommutatsionnymi sootnosheniyami dlya effekta Zeemana v pole Kulona–Diraka”, TMF, 142:1 (2005), 127–147 | DOI | MR | Zbl

[14] Karasev M. V., Novikova E. M., “Algebra s polinomialnymi kommutatsionnymi sootnosheniyami dlya effekta Zeemana–Shtarka v atome vodoroda”, TMF, 142:3 (2005), 530–555 | DOI | MR | Zbl

[15] Pereskokov A. V., “Asimptotika vblizi granits spektralnykh klasterov”, Mezhdunarodnaya konferentsiya, posvyaschennaya pamyati I. G. Petrovskogo, 23-e sovmestnoe zasedanie MMO i seminara im. I. G. Petrovskogo, Tezisy dokladov, Izd-vo MGU, M., 2011, 260–261

[16] Pereskokov A. V., “Asimptotika spektra i kvantovykh srednikh vblizi granits spektralnykh klasterov dlya vozmuschennogo dvumernogo ostsillyatora”, Matem. zametki, 92:4 (2012), 583–596 | DOI

[17] Babich V. M., Buldyrev V. S., Asimptoticheskie metody v zadachakh difraktsii korotkikh voln. Metod etalonnykh zadach, Nauka, M., 1972 | MR

[18] Maslov V. P., Kompleksnyi metod VKB v nelineinykh uravneniyakh, Nauka, M., 1977 | MR

[19] Karasev M. V., Maslov V. P., “Asimptoticheskoe i geometricheskoe kvantovanie”, UMN, 39:6 (1984), 115–173 | MR | Zbl

[20] Weinstein A., “Asymptotics of eigenvalues clusters for the laplasian plus a potential”, Duke Math. J., 44:4 (1977), 883–892 | DOI | MR | Zbl

[21] Karasev M. V., Novikova E. M., “Non-Lie permutation relations, coherent states, and quantum embedding”, Coherent transform, quantization, and Poisson geometry, Amer. Math. Soc. Trans. Ser. 2, 187, ed. Karasev M., Amer. Math. Soc., Providence, RI, 1998, 1–202 | MR | Zbl

[22] Beitmen G., Erdeii A., Vysshie transtsendentnye funktsii, V 2-x t., Nauka, M., 1973; 1974

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

[24] Golubev V. V., Lektsii po analiticheskoi teorii differentsialnykh uravnenii, Gostekhizdat, M.–L., 1950

[25] Slavyanov S. Yu., Lai V., Spetsialnye funktsii: Edinaya teoriya, osnovannaya na analize osobennostei, Nevskii Dialekt, SPb., 2002

[26] Dirac P. A. M., “Quantum electrodynamics”, Comm. Dublin Inst. Adv. Stud., Ser. A, 1 (1943), 1–36 | MR | Zbl

[27] Fedoryuk M. V., Asimptoticheskie metody dlya lineinykh obyknovennykh differentsialnykh uravnenii, Nauka, M., 1983 | MR | Zbl

[28] Olver F., Asimptotika i spetsialnye funktsii, Nauka, M., 1990 | MR | Zbl

[29] Lavrentev M. A., Shabat B. V., Metody teorii funktsii kompleksnogo peremennogo, Nauka, M., 1987 | MR

[30] Gradshtein I. S., Ryzhik I. M., Tablitsy integralov, summ, ryadov i proizvedenii, Nauka, M., 1975

[31] Dzhenkins Dzh., Odnolistnye funktsii i konformnye otobrazheniya, IL, M., 1962

[32] M. Abramovits, I. Stigan (red.), Spravochnik po spetsialnym funktsiyam s formulami, grafikami i matematicheskimi tablitsami, Nauka, M., 1979 | MR

[33] Fedoryuk M. V., Asimptotika: integraly i ryady, Nauka, M., 1987 | MR

[34] Braun P. A., “Metod VKB dlya trekhchlennykh rekurrentnykh sootnoshenii i kvazienergii angarmonicheskogo ostsillyatora”, TMF, 37:3 (1978), 355–370 | MR

[35] Vasileva A. B., “O sootvetstvii mezhdu nekotorymi svoistvami reshenii lineinykh raznostnykh sistem i obyknovennykh differentsialnykh uravnenii”, Trudy seminara po teorii differentsialnykh uravnenii s otklonyayuschimsya argumentom, 5, UDN, 1967, 21–44 | MR

[36] Maslov V. P., Fedoryuk M. V., Kvaziklassicheskoe priblizhenie dlya uravnenii kvantovoi mekhaniki, Nauka, M., 1976 | MR

[37] Maslov V. P., Kompleksnye markovskie tsepi i kontinualnyi integral Feinmana, Nauka, M., 1976 | MR

[38] Shiff L., Kvantovaya mekhanika, IL, M., 1959