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Наносистемы: физика, химия, математика, 2014, том 5, выпуск 5, страницы 619–625
(Mi nano892)
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On the number of eigenvalues of the family of operator matrices
M. I. Muminova, T. H. Rasulovb a Universiti Teknologi Malaysia, Faculty of Science, Departmentof Mathematical Sciences, 81310 UTM Johor Bahru, Malaysia
b Bukhara State University, Faculty of Physics and Mathematics, 11 M. Ikbol str., Bukhara, 200100, Uzbekistan
Аннотация:
We consider the family of operator matrices $H(K)$, $K\in\mathbb{T}^3:=(-\pi,\pi]^3$ acting in the direct sum of zero-, one- and two-particle subspaces of the bosonic Fock space. We find a finite set $\Lambda\subset\mathbb{T}^3$ to establish the existence of infinitely many eigenvalues of $H(K)$ for all $K\in\Lambda$ when the associated Friedrichs model has a zero energy resonance. It is found that for every $K\in\Lambda$ the number $N(K,z)$ of eigenvalues of $H(K)$ lying on the left of $z$, $z<0$, satisfies the asymptotic relation $\lim_{z\to -0}N(k,z)|\log|z||^{-1}=\mathcal{U}_0$ with $0<\mathcal{U}_0<\infty$, independently on the cardinality of $\Lambda$. Moreover, we show that for any $K\in\Lambda$ 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.
Ключевые слова:
operator matrix, bosonic Fock space, annihilation and creation operators, Friedrichs model, essential spectrum, asymptotics.
Поступила в редакцию: 03.06.2014
Образец цитирования:
M. I. Muminov, T. H. Rasulov, “On the number of eigenvalues of the family of operator matrices”, Наносистемы: физика, химия, математика, 5:5 (2014), 619–625
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/nano892 https://www.mathnet.ru/rus/nano/v5/i5/p619
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Страница аннотации: | 94 | PDF полного текста: | 40 | Список литературы: | 1 |
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