On the number of eigenvalues of the family of operator matrices
Abstract
We consider the family of operator matrices H(K), K ∈ T3 := (−π; π]3 acting in the direct sum of zero-, one- and two-particle subspaces of the bosonic Fock space. We find a finite set Λ ⊂ T3 to establish the existence of infinitely many eigenvalues of H(K) for all K ∈ Λ when the associated Friedrichs model has a zero energy resonance. It is found that for every K ∈ Λ, the number N (K, z) of eigenvalues of H(K) lying on the left of z, z < 0, satisfies the asymptotic relation lim z→−0 N (K, z)| log |z||−1 = U0 with 0 < U0 < ∞, independently on the cardinality of Λ. Moreover, we show that for any K ∈ Λ the operator H(K) has a finite number of negative eigenvalues if the associated Friedrichs model has a zero eigenvalue or a zero is the regular type point for positive definite Friedrichs model.
Keywords
operator matrix,
bosonic Fock space,
annihilation and creation operators,
Friedrichs model,
essential spectrum,
asymptotics
Supporting agencies: This work was supported by the IMU Einstein Foundation Program. T. H. Rasulov wishes to thank the Berlin Mathematical School and Weierstrass Institute for Applied Analysis and Stochastics for the invitation and hospitality.
About the Authors
M. I. Muminov
Universiti Teknologi Malaysia, Faculty of Science, Department of Mathematical Sciences
Malaysia
Malaysia
81310 UTM Johor Bahru
T. H. Rasulov
Bukhara State University, Faculty of Physics and Mathematics
Uzbekistan
Uzbekistan
11 M. Ikbol str., Bukhara, 200100
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Review
For citations:
Muminov M.I., Rasulov T.H. On the number of eigenvalues of the family of operator matrices. Nanosystems: Physics, Chemistry, Mathematics. 2014;5(5):619-625.
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