On the eigenvalues of a 2 x 2 block operator matrix

• Autorzy: Muminov M. I. , Rasulov T. H.

Identyfikatory

DOI

http://dx.doi.Org/10.7494/OpMath.2015.35.3.371

• Języki publikacji: EN

Abstrakty

EN

A 2 x 2 block operator matrix 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 H22 (the second diagonal entry of 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 H22 lying below z < 0, the following asymptotics is found [formula]. 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 H is proved.

Słowa kluczowe

EN

block operator matrix; Fock space; discrete and essential spectra; Birman-Schwinger principle; Efimov effect; discrete spectrum asymptotics; embedded eigenvalues

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2015
• Tom: Vol. 35, no. 3
• Strony: 371--395
• Opis fizyczny: Bibliogr. 27 poz.

Twórcy

autor

Muminov M. I.

• Universiti Teknologi Malaysia (UTM) Faculty of Science 81310 Skudai, Johor Bahru, Malaysia

autor

Rasulov T. H.

• Bukhara State University Faculty of Physics and Mathematics 11 M. Ikbol Str., Bukhara, 200100, Uzbekistan

Bibliografia

• [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 Schrodinger 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 Poincare, Phys. Theor. 60 (1994) 3, 253-290.
• [7] V. Efimov, Energy levels arising from resonant two-body forces 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 Schrodinger 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.E. Muminov, N.M. Aliev, Spectrum of the three-particle Schrodinger 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 Hamil-tonian in Fock space, Methods Funct. Anal. Topology 17 (2011) 1, 47-57.
• [17] S.N. Nabako, S.I. Yakovlev, The discrete Schrodinger 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 Efi-mov'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 Hamil-tonian, 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 Schrodinger operators, J. Func. Anal. 95 (1991) 2, 433-459.
• [25] G.R. Yodgorov, M.E. 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 Schrodinger 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.