NANOSYSTEMS: PHYSICS, CHEMISTRY, MATHEMATICS, 2014, 5 (5), P. 619–625
ON THE NUMBER OF EIGENVALUES OF THE FAMILY OF OPERATOR MATRICES
M. I. Muminov – Universiti Teknologi Malaysia, Faculty of Science, Department of Mathematical Sciences, 81310 UTM Johor Bahru, Malaysia; firstname.lastname@example.org
T. H. Rasulov – Bukhara State University, Faculty of Physics and Mathematics, 11 M. Ikbol str., Bukhara, 200100, Uzbekistan; email@example.com
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.