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Properties of the Radial Part of the Laplace Operator for l=1 in a Special Scalar Product

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We develop self-adjoint extensions of the radial part of the Laplace operator for l = 1 in a special scalar product. The product arises under the passage of the standard product from ℝ3 to the set of functions parametrizing one of two components of the transverse vector field. Similar extensions are treated for the square of the inverse operator of the radial part in question. Bibliography: 8 titles.

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References

  1. B. F. Schutz, Geometrical Methods of Mathematical Physics, Cambridge Univ. Press (1982); E. L. Hill, “The theory of vector spherical harmonics”, Amer. J. Phys., 22, 211 (1954).

  2. T. A. Bolokhov, “Extensions of the quadratic form of the transverse Laplace operator,” Zap. Nauchn. Semin. POMI, 433, 78 (2015).

    Google Scholar 

  3. K. Friedrichs, “Spektraltheorie halbbeschr¨ankter Operatoren,” Math. Ann., 109, 465–487 (1934); M. Stone, in: Linear Transformations in Hilbert spaces and Their Applications in Analysis, Amer. Math. Soc. Colloquim Publication 15, Providence (1932); or see Theorem X.23 of [7].

  4. F. A. Berezin and L. D. Faddeev, “A remark on Schrodinger’s equation with a singular potential,” Dokl. Akad. Nauk SSSR, Ser. Fiz., 137, 1011 (1961).

    MathSciNet  MATH  Google Scholar 

  5. J. Blank, P. Exner, and M. Havlcek, Hilbert Space Operators in Quantum Physics, Springer (2008).

  6. M. Reed and B. Simon, Methods of Modern Mathematical Physics. I. Functional Analysis, Academic Press, New York, London (1972).

    MATH  Google Scholar 

  7. M. Reed and B. Simon, Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-adjointness, Academic Press, New York, San Francisco, London (1975).

    MATH  Google Scholar 

  8. S. Albeverio and P. Kurasov, Singular Perturbation of Differential Operators. Solvable Schrödinger Type Operators, Cambridge Univ. Press (2000). 573

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Authors and Affiliations

  1. St.Petersburg Department of the Steklov Mathematical Institute, St.Petersburg, Russia

    T. A. Bolokhov

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  1. T. A. Bolokhov
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Correspondence to T. A. Bolokhov.

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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 434, 2015, pp. 32–52.

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Bolokhov, T.A. Properties of the Radial Part of the Laplace Operator for l=1 in a Special Scalar Product. J Math Sci 215, 560–573 (2016). https://doi.org/10.1007/s10958-016-2861-7

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  • DOI: https://doi.org/10.1007/s10958-016-2861-7

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