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Register a new userOn the instability of the essential spectrum for block Jacobi matrices
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We are interested in the phenomenon of the essential spectrum instability for a class of unbounded (block) Jacobi matrices. We give a series of sufficient conditions for the matrices from certain classes to have a discrete spectrum on a half-axis of a real line. An extensive list of examples showing the sharpness of obtained results is provided.
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We consider the family of $3 times 3$ operator matrices $H(K),$ $K in {Bbb T}^{rm d}:=(-pi; pi]^{rm d}$ arising in the spectral analysis of the energy operator of the spin-boson model of radioactive decay with two bosons on the torus ${Bbb T}^{rm d}.$ We obtain an analogue of the Faddeev equation for the eigenfunctions of $H(K)$. An analytic description of the essential spectrum of $H(K)$ is established. Further, it is shown that the essential spectrum of $H(K)$ consists the union of at most three bounded closed intervals.
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