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On the number of eigenvalues of the family of operator matrices

Аннотация:

We consider the family of operator matrices H(K), K ∈ T 3 := (−π; π] 3 acting in the direct sum of zero-, oneand two-particle subspaces of the bosonic Fock space. We find a finite set Λ ⊂ T 3 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.

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