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Izv. Akad. Nauk SSSR Ser. Mat., 1990, Volume 54, Issue 2, Pages 275–302 (Mi izv1094)  

This article is cited in 24 scientific papers (total in 26 papers)

Spectral theory in $p$-adic quantum mechanics, and representation theory

V. S. Vladimirov, I. V. Volovich, E. I. Zelenov


Abstract: Spectral theory is discussed for a harmonic oscillator in $p$-adic quantum mechanics. The problem of the decomposition into irreducible representations of the restriction of a projective representation of the symplectic group to a compact abelian subgroup is solved, the dimensions of the invariant subspaces are calculated, and the eigenfunctions are analyzed. Spectral problems are considered for $p$-adic pseudodifferential operators of Schrödinger type. A complete orthonormal system of eigenfunctions is constructed for the $p$-adic analog of the differentiation operator.

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English version:
Mathematics of the USSR-Izvestiya, 1991, 36:2, 281–309

Bibliographic databases:

Document Type: Article
UDC: 513.015.7, 512.625.5
MSC: Primary 35P05, 81C40, 81C99; Secondary 46P05
Received: 12.10.1989

Citation: V. S. Vladimirov, I. V. Volovich, E. I. Zelenov, “Spectral theory in $p$-adic quantum mechanics, and representation theory”, Izv. Akad. Nauk SSSR Ser. Mat., 54:2 (1990), 275–302; Math. USSR-Izv., 36:2 (1991), 281–309

Citation in format AMSBIB
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\by V.~S.~Vladimirov, I.~V.~Volovich, E.~I.~Zelenov
\paper Spectral theory in $p$-adic quantum mechanics, and representation theory
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1990
\vol 54
\issue 2
\pages 275--302
\mathnet{http://mi.mathnet.ru/izv1094}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1062514}
\zmath{https://zbmath.org/?q=an:0715.22029|0709.22010}
\adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?1991IzMat..36..281V}
\transl
\jour Math. USSR-Izv.
\yr 1991
\vol 36
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\pages 281--309
\crossref{https://doi.org/10.1070/IM1991v036n02ABEH002022}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. A. Yu. Khrennikov, “Mathematical methods of non-Archimedean physics”, Russian Math. Surveys, 45:4 (1990), 87–125  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    2. E. I. Zelenov, “$p$-Adic quantum mechanics and coherent states”, Theoret. and Math. Phys., 86:2 (1991), 143–151  mathnet  crossref  mathscinet  zmath  isi
    3. A. N. Kochubei, “Schrödinger-type operator over $p$-adic number field”, Theoret. and Math. Phys., 86:3 (1991), 221–228  mathnet  crossref  mathscinet  zmath  isi
    4. E. I. Zelenov, “$p$-adic quantum mechanics and coherent states 2. Oscillator eigenfunctions”, Theoret. and Math. Phys., 86:3 (1991), 258–265  mathnet  crossref  mathscinet  zmath  isi
    5. A. Kh. Bikulov, “Investigation of $p$-adic Green's function”, Theoret. and Math. Phys., 87:3 (1991), 600–610  mathnet  crossref  mathscinet  zmath  isi
    6. R. S. Ismagilov, “Spectrum of a self-adjoint operator in $L_2(K)$, where $K$ is a local field; analog of the Feynman–Kac formula”, Theoret. and Math. Phys., 89:1 (1991), 1024–1028  mathnet  crossref  mathscinet  zmath  isi
    7. V. D. Ivashchuk, “Tensor Banach algebras of projective type. I”, Theoret. and Math. Phys., 91:1 (1992), 336–345  mathnet  crossref  mathscinet  zmath  isi
    8. A. N. Kochubei, “The differentiation operator on subsets of the field of $p$-adic numbers”, Russian Acad. Sci. Izv. Math., 41:2 (1993), 289–305  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    9. V. S. Vladimirov, “On spectral properties of $p$-adic pseudodifferential operators of Schrödinger type”, Russian Acad. Sci. Izv. Math., 41:1 (1993), 55–73  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    10. Anatoly N. Kochubei, “A Schrödinger-type equation over the field of p-adic numbers”, J Math Phys (N Y ), 34:8 (1993), 3420  crossref  mathscinet  isi
    11. E. I. Zelenov, “The infinite-demensional $p$-adic symplectic group”, Russian Acad. Sci. Izv. Math., 43:3 (1994), 421–441  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    12. A. Yu. Khrennikov, “$p$-Adic probability theory and its applications. The principle of statistical stabilization of frequencies”, Theoret. and Math. Phys., 97:3 (1993), 1340–1348  mathnet  crossref  mathscinet  zmath  isi
    13. A. A. Gonchar, G. I. Marchuk, S. P. Novikov, “Vasilii Sergeevich Vladimirov (on his seventieth birthday)”, Russian Math. Surveys, 48:1 (1993), 201–212  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    14. B. G. Dragovich, “Adelic model of harmonic oscillator”, Theoret. and Math. Phys., 101:3 (1994), 1404–1412  mathnet  crossref  mathscinet  zmath  isi
    15. A. N. Kochubei, “Gaussian integrals and spectral theory over a local field”, Russian Acad. Sci. Izv. Math., 45:3 (1995), 495–503  mathnet  crossref  mathscinet  zmath  isi
    16. V.S. Vladimirov, “Some problems of analysis on the field ofp-adic numbers”, Integral Transforms and Special Functions, 6:1-4 (1998), 111  crossref
    17. Lee-Chae Jang, Min-Soo Kim, Jin-Woo Son, Taekyun Kim, Seog-Hoon Rim, “On p-adic Bounded Functions, II”, Journal of Mathematical Analysis and Applications, 264:1 (2001), 21  crossref
    18. Denis Krutikov, “Spectra of <i>p</i>-adic Schr dinger-type operators with random radial potentials”, J Phys A Math Gen, 36:15 (2003), 4433  crossref  mathscinet  zmath  isi  elib
    19. V. S. Vladimirov, “Tables of Integrals of Complex-Valued Functions of $p$-Adic Arguments”, Proc. Steklov Inst. Math., 284, suppl. 2 (2014), S1–S59  mathnet  crossref  crossref  mathscinet  zmath  isi
    20. M. K. Kerimov, “Vasiliĭ Sergeevich Vladimirov (on the occasion of his eightieth birthday)”, Comput. Math. Math. Phys., 43:11 (2003), 1541–1549  mathnet  mathscinet
    21. S. V. Kozyrev, “Methods and Applications of Ultrametric and $p$-Adic Analysis: From Wavelet Theory to Biophysics”, Proc. Steklov Inst. Math., 274, suppl. 1 (2011), S1–S84  mathnet  crossref  crossref  zmath  isi  elib
    22. S. V. Kozyrev, A. Yu. Khrennikov, V. M. Shelkovich, “$p$-Adic wavelets and their applications”, Proc. Steklov Inst. Math., 285 (2014), 157–196  mathnet  crossref  crossref  isi
    23. R. S. Ismagilov, “Asymptotic form of the spectrum of operators associated with $p$-adic fields”, Theoret. and Math. Phys., 180:1 (2014), 753–758  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    24. R. S. Ismagilov, “Spectral Trace Formula for Local Fields”, Math. Notes, 98:6 (2015), 926–931  mathnet  crossref  crossref  mathscinet  isi  elib
    25. Dragovich B. Khrennikov A.Yu. Kozyrev S.V. Volovich I.V. Zelenov E.I., “P-Adic Mathematical Physics: the First 30 Years”, P-Adic Numbers Ultrametric Anal. Appl., 9:2 (2017), 87–121  crossref  isi
    26. Saberi I., “Holography and Local Fields”, P-Adic Numbers Ultrametric Anal. Appl., 10:3 (2018), 151–165  crossref  isi
  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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