Muminov, Mukhiddin and Rasulov, T. H.
(2014)
*On the number of eigenvalues of the family of operator matrices.*
Nanosystems: Physics, Chemistry, Mathematics, 5
(5).
pp. 619-625.
ISSN 2220-8054

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## 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.

Item Type: | Article |
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Uncontrolled Keywords: | annihilation and creation operators, Friedrichs model, essential spectrum, asymptotics |

Subjects: | Q Science > QA Mathematics |

Divisions: | Science |

ID Code: | 59855 |

Deposited By: | Haliza Zainal |

Deposited On: | 23 Jan 2017 00:24 |

Last Modified: | 24 Apr 2022 04:50 |

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