Infinite number of eigenvalues of $2\times 2$ operator matrices: Asymptotic discrete spectrum

T. H. Rasulov ; E. B. Dilmurodov

Teoretičeskaâ i matematičeskaâ fizika, Tome 205 (2020) no. 3, pp. 368-390

Résumé

We study an unbounded $2\times2$ operator matrix $\mathcal{A}$ in the direct product of two Hilbert spaces. We obtain asymptotic formulas for the number of eigenvalues of $\mathcal{A}$. We consider a $2\times2$ operator matrix $\mathcal{A}_\mu$, where $\mu>0$ is the coupling constant, associated with the Hamiltonian of a system with at most three particles on the lattice $\mathbb{Z}^3$. We find the critical value $\mu_0$ of the coupling constant $\mu$ for which $\mathcal{A}_{\mu_0}$ has an infinite number of eigenvalues. These eigenvalues accumulate at the lower and upper bounds of the essential spectrum. We obtain an asymptotic formula for the number of such eigenvalues in both the left and right parts of the essential spectrum.

Détail

Keywords: operator matrix, coupling constant, dispersion function, Fock space, creation operator, annihilation operator, Birman–Schwinger principle, essential spectrum, discrete spectrum, asymptotics.

Bibliographie

[1] C. Tretter, Spectral Theory of Block Operator Matrices and Applications, Imperial College Press, London, 2008 | DOI | MR

[2] D. Mattis, “The few-body problem on lattice”, Rev. Modern Phys., 58:2 (1986), 361–379 | DOI | MR

[3] K. O. Fridrikhs, Vozmuschenie spektra operatorov v gilbertovom prostranstve, Mir, M., 1969

[4] V. A. Malyshev, R. A. Minlos, “Klastepnye opepatopy”, Tr. sem. im. I. G. Petrovskogo, 1983, no. 9, 63–80 | MR | Zbl

[5] S. Albeverio, S. N. Lakaev, T. H. Rasulov, “On the spectrum of an Hamiltonian in Fock space. Discrete spectrum asymptotics”, J. Stat. Phys., 127:2 (2007), 191–220 | DOI | MR

[6] S. Albeverio, S. N. Lakaev, T. H. Rasulov, “The Efimov effect for a model operator associated with the Hamiltonian of a non conserved number of particles”, Methods Func. Anal. Topol., 13:1 (2007), 1–16 | MR

[7] S. N. Lakaev, T. Kh. Rasulov, “Ob effekte Efimova v modeli teorii vozmuschenii suschestvennogo spektra”, Funkts. analiz i ego pril., 37:1 (2003), 81–84 | DOI | DOI | MR | Zbl

[8] V. N. Efimov, “Slabosvyazannye sostoyaniya trekh rezonansno vzaimodeistvuyuschikh chastits”, YaF, 12:5 (1970), 1080–1091

[9] D. R. Yafaev, “K teorii diskretnogo spektra trekhchastichnogo operatora Shredingera”, Matem. sb., 94(136):4(8) (1974), 567–593 | DOI | MR | Zbl

[10] A. V. Sobolev, “The Efimov effect. Discrete spectrum asymptotics”, Commun. Math. Phys., 156:1 (1993), 101–126 | DOI | MR

[11] H. Tamura, “The Efimov effect of three-body Schrödinger operators: asymptotics for the number of negative eigenvalues”, Nagoya Math. J., 130 (1993), 55–83 | DOI | MR

[12] S. N. Lakaev, “O beskonechnom chisle tpekhchastichnykh svyazannykh sostoyanii sistemy tpekh kvantovykh peshetchatykh chastits”, TMF, 89:1 (1991), 94–104 | DOI | MR

[13] S. N. Lakaev, “Ob effekte Efimova v sisteme tpekh odinakovykh kvantovykh chastits”, Funkts. analiz i ego ppil., 27:3 (1993), 15–28 | DOI | MR | Zbl

[14] S. N. Lakaev, M. E. Muminov, “Suschestvennyi i diskretnyi spektr trekhchastichnogo operatora Shredingera na reshetke”, TMF, 135:3 (2003), 478–503 | DOI | DOI | MR | Zbl

[15] Zh. I. Abdullaev, S. N. Lakaev, “Asimptotika diskretnogo spektra raznostnogo trekhchastichnogo operatora Shredingera na reshetke”, TMF, 136:2 (2003), 231–245 | DOI | DOI | MR | Zbl

[16] S. Albeverio, S. N. Lakaev, Z. I. Muminov, “Schrödinger operators on lattices. The Efimov effect and discrete spectrum asymptotics”, Ann. Henri Poincaré, 5:4 (2004), 743–772, arXiv: math-ph/0312026 | DOI | MR

[17] S. N. Lakaev, Z. E. Muminov, “Asimptotika dlya chisla sobstvennykh znachenii trekhchastichnogo operatora Shredingera na reshetke”, Funk. analiz i ego pril., 37:3 (2003), 80–84 | DOI | DOI | MR | Zbl

[18] M. I. Muminov, T. H. Rasulov, “On the eigenvalues of a $2\times 2$ block operator matrix”, Opuscula Math., 35:3 (2015), 371–395 | DOI | MR

[19] M. I. Muminov, T. H. Rasulov, “Infiniteness of the number of eigenvalues embedded in the essential spectrum of a $2\times 2$ operator matrix”, Eurasian Math. J., 5:2 (2014), 60–77 | MR

[20] M. I. Muminov, T. H. Rasulov, “Embedded eigenvalues of an Hamiltonian in bosonic Fock space”, Commun. Math. Anal., 17:1 (2014), 1–22 | MR | Zbl

[21] 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, American Mathematical Society Translations: Ser. 2, 177, eds. R. L. Dobrushin, R. A. Minlos, M. A. Shubin, A. M. Vershik, AMS, Providence, RI, 1996, 159–193 | DOI | MR

[22] Yu. V. Zhukov, R. A. Minlos, “Spektr i rasseyanie v modeli “spin-bozon” s ne bolee chem tremya fotonami”, TMF, 103:1 (1995), 63–81 | DOI | MR | Zbl

[23] M. Rid, B. Saimon, Metody sovremennoi matematicheskoi fiziki, v. 4, Analiz operatorov, Mir, M., 1982 | MR | MR | Zbl | Zbl

[24] M. Sh. Birman, M. Z. Salomyak, Spektralnaya teoriya samosopryazhennykh operatorov v gilbertovom prostranstve, Izd-vo Leningr. un-ta, L., 1980

[25] R. P. Feynman, “Creation and annihilation operators”, Statistical Mechanics: A Set of Lectures, CRC Press, London, 2018, 151–197

[26] M. Muminov, H. Neidhardt, T. Rasulov, “On the spectrum of the lattice spin-boson Hamiltonian for any coupling: 1D case”, J. Math. Phys., 56:5 (2015), 053507, 24 pp., arXiv: 1410.4763 | DOI | MR

[27] T. Kh. Rasulov, “O vetvyakh suschestvennogo spektra reshetchatoi modeli spin-bozona s ne bolee chem dvumya fotonami”, TMF, 186:2 (2016), 293–310 | DOI | DOI | MR