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Complex Analysis and Operator Theory

, Volume 13, Issue 3, pp 1313–1323 | Cite as

Spectral Attributes of Self-Adjoint Fredholm Operators in Hilbert Space: A Rudimentary Insight

  • Mukhiddin I. MuminovEmail author
  • S. K. Ghoshal
Article
  • 42 Downloads

Abstract

In defining the finiteness or infiniteness conditions of discrete spectrum of the Schrodinger operators, a fundamental understanding on \(n(1, F(\cdot ))\) is crucial, where n(1, F) is the number of eigenvalues of the Fredholm operator F to the right of 1. Driven by this idea, this paper provided the invertibility condition for some class of operators. A sufficient condition for finiteness of the discrete spectrum involving the self-adjoint operator acting on Hilbert space was achieved. A relation was established between the eigenvalue 1 of the self-adjoint Fredholm operator valued function \(F(\cdot )\) defined in the interval of (ab) and discontinuous points of the function \(n(1, F(\cdot ))\). Besides, the obtained relation allowed us to define the finiteness of the numbers \(z\in (a,b)\) for which 1 is an eigenvalue of F(z) even if \(F(\cdot )\) is not defined at a and b. Results were validated through some examples.

Keywords

Fredholm analytic theorem Self-adjoint operator Discrete spectrum Essential spectrum 

Mathematics Subject Classification

47Axx 35Pxx 

References

  1. 1.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics. Analysis of Operators, vol. 4. Academic Press, London (1980)zbMATHGoogle Scholar
  2. 2.
    Birman, M.Sh., Solomyak, M.Z.: Spectral Theory of Selfadjoint Operators in Hubert Space. D. Reidel, Dordrecht (1987)Google Scholar
  3. 3.
    Faddeev, L.D.: Mathematical Aspects of the Three-Body Problem in the Quantum Scattering Theory. Trudy Matematicheskogo Instituta Imeni V.A. Steklova, vol. 69. Israel Program for Scientific Translations, Jerusalem (1963). (Russian)Google Scholar
  4. 4.
    Faddeev, L.D., Merkuriev, S.P.: Quantum Scattering Theory for Several Particle Systems. Kluwer Academic Publishers, Boston (1993)CrossRefzbMATHGoogle Scholar
  5. 5.
    Yafaev, D.R.: On the theory of the discrete spectrum of the three-particle Schrödinger operator. Math. USSR-Sb. 23, 535–559 (1974)CrossRefzbMATHGoogle Scholar
  6. 6.
    Sobolev, A.V.: The Efimov effect. Discrete spectrum asymptotics. Commun. Math. Phys. 156(1), 101–126 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Tamura, H.: The Efimov effect of three-body Schrödinger operator. J. Funct. Anal. 95, 433–459 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Muminov, M.É.: The infiniteness of the number of eigenvalues in the gap in the essential spectrum for the three-particle Schrödinger operator on a lattice. Theor. Math. Phys. 159(2), 299–317 (2009)CrossRefzbMATHGoogle Scholar
  9. 9.
    Lakaev, S.N., Muminov, M.I.: Essential and discrete spectra of the three-particle Schrödinger operator on a lattice. Teoret. Mat. Fiz. 135(3), 478–503 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Muminov, M.I.: Formula for the number of eigenvalues of a three-particle Schrödinger operator on a lattice. Teoret. Mat. Fiz. 164(1), 46–61 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Abdullaev, J.I., Ikromov, I.A.: Finiteness of the number of eigenvalues of the two-particle Schrödinger operator on a lattice. Theor. Math. Phys. 152(3), 1299–1312 (2007)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of Mathematical of Sciences, Faculty of ScienceUniversiti Teknologi Malaysia (UTM)Johor BahruMalaysia
  2. 2.Department of Physics, Faculty of ScienceUniversiti Teknologi Malaysia (UTM)Johor BahruMalaysia

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