Analysis of the spectrum of a 2×2 operator matrix. Discrete spectrum asymptotics
https://doi.org/10.17586/2220-8054-2020-11-2-138-144
Abstract
We consider a 2×2 operator matrix Aµ, µ > 0 related with the lattice systems describing two identical bosons and one particle, another nature in interactions, without conservation of the number of particles. We obtain an analog of the Faddeev equation and its symmetric version for the eigenfunctions of Aµ. We describe the new branches of the essential spectrum of Aµ via the spectrum of a family of generalized Friedrichs models. It is established that the essential spectrum of Aµ consists the union of at most three bounded closed intervals and their location is studied. For the critical value µ0 of the coupling constant µ we establish the existence of infinitely many eigenvalues, which are located in the both sides of the essential spectrum of Aµ. In this case, an asymptotic formula for the discrete spectrum of Aµ is found.
Keywords
operator matrix,
bosonic Fock space,
coupling constant,
dispersion function,
essential and discrete spectrum,
Birman–Schwinger principle,
spectral subspace,
Weyl creterion
Supporting agencies: The authors thank the anonymous referee for reading the manuscript carefully and for making valuable suggestions.
About the Authors
T. H. Rasulov
Faculty of Physics and Mathematics, Bukhara State University
Uzbekistan
Uzbekistan
M.Ikbol str. 11, 200100 Bukhara
E. B. Dilmurodov
Faculty of Physics and Mathematics, Bukhara State University
Uzbekistan
Uzbekistan
M.Ikbol str. 11, 200100 Bukhara
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Review
For citations:
Rasulov T.H., Dilmurodov E.B. Analysis of the spectrum of a 2×2 operator matrix. Discrete spectrum asymptotics. Nanosystems: Physics, Chemistry, Mathematics. 2020;11(2):138–144. https://doi.org/10.17586/2220-8054-2020-11-2-138-144
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