Geodesic
$p$-Adic and adelic harmonic oscillator with a time-dependent frequency
Teoretičeskaâ i matematičeskaâ fizika, Tome 124 (2000) no. 2, pp. 239-248 
G. S. Djordjevič; B. G. Dragovich. $p$-Adic and adelic harmonic oscillator with a time-dependent frequency. Teoretičeskaâ i matematičeskaâ fizika, Tome 124 (2000) no. 2, pp. 239-248. http://geodesic.mathdoc.fr/item/TMF_2000_124_2_a3/
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     author = {G. S. Djordjevi\v{c} and B. G. Dragovich},
     title = {$p${-Adic} and adelic harmonic oscillator with a time-dependent frequency},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {239--248},
     year = {2000},
     volume = {124},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2000_124_2_a3/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

The classical and quantum formalism for a $p$-adic and adelic harmonic oscillator with a time-dependent frequency is developed, and general formulas are obtained for the main theoretical quantities. In particular, the $p$-adic propagator is calculated, and the existence of a simple vacuum state as well as adelic quantum dynamics is shown. A spatial discreteness and a $p$-adic quantum mechanical phase are noted.

[1] W. H. Schikhof, Ultrametric Calculus, Cambridge Univ. Press, Cambridge, 1984 | MR | Zbl

[2] L. Brekke, P. G. O. Freund, Phys. Rep., 233 (1993), 1 | DOI | MR

[3] V. S. Vladimirov, I. V. Volovich, E. I. Zelenov, $p$-Adicheskii analiz i matematicheskaya fizika, Fiz.-mat. lit.; Nauka, M., 1994 | MR | Zbl

[4] A. Khrennikov, $p$-Adic Valued Distributions in Mathematical Physics, Kluwer Acad. Publ., Dordrecht, 1994 | MR | Zbl

[5] A. Khrennikov, Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models, Kluwer Acad. Publ., Dordrecht, 1997 | MR | Zbl

[6] I. M. Gelfand, M. I. Graev, I. I. Pyatetskii-Shapiro, Teoriya predstavlenii i avtomorfnye funktsii, Nauka, M., 1966 ; A. Weil, Adeles and Algebraic Groups, Birkhäuser, Boston, 1982 | MR | MR | Zbl

[7] V. S. Vladimirov, I. V. Volovich, Commun. Math. Phys., 123 (1989), 659 | DOI | MR | Zbl

[8] Ph. Ruelle, E. Thiran, D. Verstegen, J. Weyers, J. Math. Phys., 30 (1989), 2854 | DOI | MR | Zbl

[9] B. Dragovich, TMF, 101:3 (1994), 349 ; B. Dragovich, Int. J. Mod. Phys. A, 10 (1995), 2349 | MR | Zbl | DOI | MR | Zbl

[10] B. Dragovich, “Adelic Wave Function of the Universe”, Proc. Third Friedmann Intern. Seminar on Gravitation and Cosmology, eds. Yu. N. Gnedin, A. A. Grib, and V. M. Mostepanenko, Friedmann Lab. Publ., St. Petersburg, 1995, 311

[11] G. S. Djordjević, B. Dragovich, Lj Nešić, Mod. Phys. Lett. A, 14 (1999), 317 | DOI | MR

[12] A. D. Jannussis, B. S. Bartzis, Phys. Lett. A, 129 (1988), 263 ; R. J. Glauber, “Quantum theory of particle trapping by oscillating fields”, Quantum Measurements in Optics, eds. P. Tombesi and D. F. Walls, Plenum Press, N. Y., 1992, 3 ; V. V. Dodonov, O. V. Man'ko, V. I. Man'ko, Phys. Lett. A, 175 (1993), 1 | DOI | MR | DOI | DOI | MR

[13] J. J. Halliwell, S. W. Hawking, Phys. Rev. D, 31 (1985), 1777 | DOI | MR

[14] D. C. Khandekar, S. V. Lawande, J. Math. Phys., 16 (1975), 384 ; C. P. Natividade, Am. J. Phys., 56 (1988), 921 ; C. Farina, A. J. Segui-Santonja, Phys. Lett. A, 184 (1993), 23 | DOI | DOI | DOI | MR | Zbl

[15] G. S. Djordjević, B. Dragovich, Mod. Phys. Lett. A, 12 (1997), 1455 | DOI | MR | Zbl

[16] B. Dragovich, Integral Transforms and Special Functions, 6 (1998), 197 | DOI | MR | Zbl