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New Results in Operator Theory and Its Applications pp 163–185Cite as

Birkhäuser

Commuting Nonselfadjoint Operators and a Unified Theory of Waves and Corpuscles

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Part of the book series: Operator Theory: Advances and Applications ((OT,volume 98))

Abstract

In this paper, which is closely connected with our previous study ”What is a particle from the standpoint of systems theory?” [3], we consider the case of pairs of commuting operators with one-dimensional imaginary parts and we develop a unified theory of waves and particles for the case of Dirac equations. Particles emerge in Space as localized fields with singularities at their centers and wave equations appear as compatibility conditions for an overdetermined pair of equations of corresponding open systems.

To Israel Glazman, my dearest friend from the very childhood. May his memory live for ever.

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References

  1. N.I. Akhieser and I.M. Glazman: Theory of Linear Operators in Hilbert Space, Pitman Advanced Publishing Program, London, 1980.

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  2. Louis de Broglie: Non-Linear Wave Mechanics. Elsevier, Amsterdam, 1960.

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  3. M.S. Livšic: What is a Particle from the Standpoint of Systems Theory?, Integral Equations Operator Theory, 14 (1991), 552–563.

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  4. M.S. Livsic: Commuting Nonselfadjoint Operators and Collective Motions of Systems, in ”Commuting Nonselfadjoint Operators in Hilbert Space”, Lecture Notes in Math., 1272, Springer-Verlag, 1987.

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  5. M.S. Livsic and A.A. Jancevich: Theory of Operator Colligations in Hilbert Space, J. Wiley, New York, 1979.

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  6. Nicholas Rescher: G.W.Leibniz’s Monadology (an edition for students), London, 1991.

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Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Sciences, Ben-Gurion University of the Negev, Beer-Sheva, 84105, Israel

    M. S. Livšic

Authors
  1. M. S. Livšic
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Editor information

Editors and Affiliations

  1. Raymond and Beverly Sackler Faculty of Exact Sciences School of Mathematical Sciences, Tel Aviv University, IL-69978, Tel Aviv, Israel

    I. Gohberg  & Yu. Lyubich  & 

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© 1997 Springer Basel AG

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Livšic, M.S. (1997). Commuting Nonselfadjoint Operators and a Unified Theory of Waves and Corpuscles. In: Gohberg, I., Lyubich, Y. (eds) New Results in Operator Theory and Its Applications. Operator Theory: Advances and Applications, vol 98. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8910-0_12

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  • DOI: https://doi.org/10.1007/978-3-0348-8910-0_12

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9824-9

  • Online ISBN: 978-3-0348-8910-0

  • eBook Packages: Springer Book Archive

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