Abstract
We study the spectrum of an operator matrix arising in the spectral analysis of the energy operator of the spin-boson model of radioactive decay with two bosons on the torus. An analytic description of the essential spectrum is established. Further, a criterion for the finiteness of the number of eigenvalues below the bottom of the essential spectrum is derived.
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Adamyan, V., Mennicken, R., and Saurer, J. On the discrete spectrum of some selfadjoint operator matrices. J. Operator Theory 39, 1 (1998), 3–41.
Albeverio, S., Lakaev, S.N., and Rasulov, T.H. On the spectrum of a Hamiltonian in Fock space. Discrete spectrum asymptotics. J. Stat. Phys. 127, 2 (2007), 191–220.
Atkinson, F.V., Langer, H., Mennicken, R., and Shkalikov, A.A. The essential spectrum of some matrix operators. Math. Nachr. 167 (1994), 5–20.
Birman, M.S., and Solomjak, M.Z. Spectral theory of selfadjoint operators in Hilbert space. Mathematics and its Applications (Soviet Series). D. Reidel Publishing Co., Dordrecht, 1987. Translated from the 1980 Russian original by S. Khrushchëv and V. Peller.
Hübner, M., and Spohn, H. Atom interacting with photons: an N-body Schrödinger problem. Tech. rep., 1994.
Hübner, M., and Spohn, H. Spectral properties of the spin-boson Hamiltonian. Ann. Inst. H. Poincaré Phys. Théor. 62, 3 (1995), 289–323.
Kraus, M., Langer, M., and Tretter, C. Variational principles and eigenvalue estimates for unbounded block operator matrices and applications. Journal of Computational and Applied Mathematics 171, 1-2 (2004), 311–334. Special issue on the occasion of the eightieth birthday of Prof. W.M. Everitt.
Lakaev, S.N., and Rasulov, T.K. A model in the theory of perturbations of the essential spectrum of many-particle operators. Mat. Zametki 73, 4 (2003), 556–564.
Langer, H., Langer, M., and Tretter, C. Variational principles for eigenvalues of block operator matrices. Indiana University Mathematics Journal 51, 6 (2002), 1427–1460.
Marletta, M., and Tretter, C. Essential spectra of coupled systems of differential equations and applications in hydrodynamics. J. Differential Equations 243, 1 (2007), 36–69.
Muminov, M., Neidhardt, H., and Rasulov, T. On the spectrum of the lattice spin-boson Hamiltonian for any coupling: 1D case. J. Math. Phys. 56, 5 (2015), 053507, 24.
Rasulov, T.K. On branches of the essential spectrum of the lattice spin-boson model with at most two photons. Teoret. Mat. Fiz. 186, 2 (2016), 293–310.
Reed, M., and Simon, B. Methods of modern mathematical physics. IV. Analysis of operators. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978.
Sobolev, A.V. The Efimov effect. Discrete spectrum asymptotics. Comm. Math. Phys. 156, 1 (1993), 101–126.
Tretter, C. Spectral theory of block operator matrices and applications. Imperial College Press, London, 2008.
Zhukov, Y.V., and Minlos, R.A. The spectrum and scattering in the “spin-boson” model with at most three photons. Teoret. Mat. Fiz. 103, no. 1 (1995), 63–81.
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Ibrogimov, O.O., Tretter, C. (2018). On the Spectrum of an Operator in Truncated Fock Space. In: Alpay, D., Kirstein, B. (eds) Indefinite Inner Product Spaces, Schur Analysis, and Differential Equations. Operator Theory: Advances and Applications(), vol 263. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-68849-7_12
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DOI: https://doi.org/10.1007/978-3-319-68849-7_12
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