Opuscula Math. 35, no. 3 (2015), 371-395

Opuscula Mathematica

On the eigenvalues of a 2ˣ2 block operator matrix

Mukhiddin I. Muminov
Tulkin H. Rasulov

Abstract. A \(2\times2\) block operator matrix \({\mathbf H}\) acting in the direct sum of one- and two-particle subspaces of a Fock space is considered. The existence of infinitely many negative eigenvalues of \(H_{22}\) (the second diagonal entry of \({\bf H}\)) is proved for the case where both of the associated Friedrichs models have a zero energy resonance. For the number \(N(z)\) of eigenvalues of \(H_{22}\) lying below \(z\lt0\), the following asymptotics is found \[\lim\limits_{z\to -0} N(z) |\log|z||^{-1}=\,{\mathcal U}_0 \quad (0\lt {\mathcal U}_0\lt \infty).\] Under some natural conditions the infiniteness of the number of eigenvalues located respectively inside, in the gap, and below the bottom of the essential spectrum of \({\mathbf H}\) is proved.

Keywords: block operator matrix, Fock space, discrete and essential spectra, Birman-Schwinger principle, the Efimov effect, discrete spectrum asymptotics, embedded eigenvalues.

Mathematics Subject Classification: 81Q10, 35P20, 47N50.

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  1. S. Albeverio, On bound states in the continuum of \(N\)-body systems and the Virial theorem, Ann. Phys. 1 (1972), 167-176.
  2. S. Albeverio, S.N. Lakaev, R.Kh. Djumanova, The essential and discrete spectrum of a model operator associated to a system of three identical quantum particles, Rep. Math. Phys. 63 (2009) 3, 359-380.
  3. S. Albeverio, S.N. Lakaev, Z.I. Muminov, On the number of eigenvalues of a model operator associated to a system of three-particles on lattices, Russian J. Math. Phys. 14 (2007) 4, 377-387.
  4. R.D. Amado, J.V. Noble, On Efimov's effect: a new pathology of three-particle systems, Phys. Lett. B. 35 (1971), 25-27; II. Phys. Lett. D. 5 (1972) 3, 1992-2002.
  5. F.A. Berezin, M.A. Shubin, The Schrödinger Equation, Kluwer Academic Publishers, Dordrecht/Boston/London, 1991.
  6. G.F. Dell'Antonio, R. Figari, A. Teta, Hamiltonians for systems of \(N\) particles interacting through point interactions, Ann. Inst. Henri Poincaré, Phys. Theor. 60 (1994) 3, 253-290.
  7. V. Efimov, Energy levels arising from resonant two-bodyforces in a three-body system, Phys. Lett. B 33 (1970) 8, 563-564.
  8. K.O. Friedrichs, Perturbation of Spectra in Hilbert Space, Amer. Math. Soc., Providence, Rhole Island, 1965.
  9. I.M. Glazman, Direct Methods of the Qualitative Spectral Analysis of Singular Differential Operators, J.: IPS Trans., 1965.
  10. P.R. Halmos, A Hilbert Space Problem Book, Springer-Verlag New York Inc., 2nd ed., 1982.
  11. V.A. Malishev, R.A. Minlos, Linear Infinite-particle Operators, Translations of Mathematical Monographs, 143, AMS, Providence, RI, 1995.
  12. R.A. Minlos, H. Spohn, The three-body problem in radioactive decay: the case of one atom and at most two photons, Topics in Statistical and Theoretical Physics, Amer. Math. Soc. Transl., Ser. 2, 177 (1996), 159-193.
  13. A.I. Mogilner, Hamiltonians in solid state physics as multiparticle discrete Schrödinger operators: problems and results, Advances in Sov. Math. 5 (1991), 139-194.
  14. M.I. Muminov, Positivity of the two-particle Hamiltonian on a lattice, Teor. Mat. Fiz. 153 (2007) 3, 381-387 [in Russian]; Engl. transl. [in:] Theor. Math. Phys. 153 (2007) 3, 1671-1676.
  15. M.É. Muminov, N.M. Aliev, Spectrum of the three-particle Schrödinger operator on a one-dimensional lattice, Teor. Mat. Fiz. 171 (2012) 3, 387-403 [in Russian]; Engl. transl. [in:] Theor. Math. Phys. 171 (2012) 3, 754-768.
  16. M.I. Muminov, T.H. Rasulov, The Faddeev equation and essential spectrum of a Hamiltonian in Fock space, Methods Funct. Anal. Topology 17 (2011) 1, 47-57.
  17. S.N. Nabako, S.I. Yakovlev, The discrete Schrödinger operators. A point spectrum lying in the continuous spectrum, Algebra i Analiz 4 (1992) 3, 183-195 [in Russian]; Engl. transl. [in:] St. Petersburg Math. J. 4 (1993) 3, 559-568.
  18. Yu.N. Ovchinnikov, I.M. Sigal, Number of bound states of three-body systems and Efimov's effect, Ann. Phys. 123 (1979) 2, 274-295.
  19. T.Kh. Rasulov, The Faddeev equation and the location of the essential spectrum of a model multi-particle operator, Izvestiya VUZ. Matematika (2008) 12, 59-69 [in Russian]; Engl. transl. [in:] Russian Math. (Iz. VUZ) 52 (2008) 12, 50-59.
  20. T.Kh. Rasulov, On the structure of the essential spectrum of a model many-body Hamiltonian, Matem. Zametki 83 (2008) 1, 78-86 [in Russian]; Engl. transl. [in:] Mathem. Notes 83 (2008) 1, 80-87.
  21. T.Kh. Rasulov, Study of the essential spectrum of a matrix operator, Teor. Mat. Fiz. 164 (2010) 1, 62-77 [in Russian]; Engl. transl. in: Theor. Math. Phys. 164 (2010) 1, 883-895.
  22. M. Reed, B. Simon, Methods of Modern Mathematical Physics. IV: Analysis of Operators, Academic Press, New York, 1979.
  23. A.V. Sobolev, The Efimov effect. Discrete spectrum asymptotics, Commun. Math. Phys. 156 (1993) 1, 101-126.
  24. H. Tamura, The Efimov effect of three-body Schrödinger operators, J. Func. Anal. 95 (1991) 2, 433-459.
  25. G.R. Yodgorov, M.É. Muminov, Spectrum of a model operator in the perturbation theory of the essential spectrum, Teor. Mat. Fiz. 144 (2005) 3, 544-554 [in Russian]; Engl. transl. [in:] Theor. Math. Phys. 144 (2005) 3, 1344-1352.
  26. D.R. Yafaev, On the theory of the discrete spectrum of the three-particle Schrödinger operator, Mat. Sbornik 94(136) (1974) 4(8), 567-593 [in Russian]; Engl. transl. [in:] Math. USSR-Sb. 23 (1974) 4, 535-559.
  27. Yu. Zhukov, R. Minlos, Spectrum and scattering in a "spin-boson" model with not more than three photons, Teor. Mat. Fiz. 103 (1995) 1, 63-81 [in Russian]; Engl. transl. [in:] Theor. Math. Phys. 103 (1995) 1, 398-411.
  • Mukhiddin I. Muminov
  • Universiti Teknologi Malaysia (UTM), Faculty of Science, 81310 Skudai, Johor Bahru, Malaysia
  • Tulkin H. Rasulov
  • Bukhara State University, Faculty of Physics and Mathematics, 11 M. Ikbol Str., Bukhara, 200100, Uzbekistan
  • Communicated by P.A. Cojuhari.
  • Received: 2014-05-26.
  • Revised: 2014-08-07.
  • Accepted: 2014-09-07.
  • Published online: 2014-12-15.
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Cite this article as:
Mukhiddin I. Muminov, Tulkin H. Rasulov, On the eigenvalues of a 2ˣ2 block operator matrix, Opuscula Math. 35, no. 3 (2015), 371-395, http://dx.doi.org/10.7494/OpMath.2015.35.3.371

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