General information
Latest issue
Forthcoming papers
Impact factor
Guidelines for authors
License agreement
Submit a manuscript

Search papers
Search references

Latest issue
Current issues
Archive issues
What is RSS

Izv. RAN. Ser. Mat.:

Personal entry:
Save password
Forgotten password?

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.

Full text: PDF file (1239 kB)
References: PDF file   HTML file

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

Bibliographic databases:

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
\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
\jour Math. USSR-Izv.
\yr 1991
\vol 36
\issue 2
\pages 281--309

Linking options:

    SHARE: FaceBook Twitter Livejournal

    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 p-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  elib  elib
    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
    Number of views:
    This page:914
    Full text:233
    First page:4

    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2019