Analysis of the spectrum of a 2x2 operator matrix. Discrete spectrum asymptotics

Abstract

We consider a 2 × 2 operator matrix Aµ, µ > 0 related with the lattice systems describing two identical bosons and one particle, another nature in interactions, without conservation of the number of particles. We obtain an analog of the Faddeev equation and its symmetric version for the eigenfunctions of Aµ. We describe the new branches of the essential spectrum of Aµ via the spectrum of a family of generalized Friedrichs models. It is established that the essential spectrum of Aµ consists the union of at most three bounded closed intervals and their location is studied. For the critical value µ 0 of the coupling constant µ we establish the existence of infinitely many eigenvalues, which are located in the both sides of the essential spectrum of Aµ. In this case, an asymptotic formula for the discrete spectrum of Aµ is found.

On the Eigenvalues of a Soluble Model of Coupled Molecular and Nuclear Hamiltonians

“…ε ρ ∈ Applying Lemma 5, we obtain that the quantity z ε is a real quantity. This, together with the relation (10), implies the inequalities…”

“…. We noticed that threshold eigenvalue, virtual level (threshold energy resonance), and threshold energy expansion for the associated Fredholm determinant of a generalized Friedrichs model with μ = 0 have been studied in [9][10][11][12]. The localization and number of discrete eigenvalues of this model are investigated in [13].…”

Existence of the eigenvalues of a tensor sum of the Friedrichs models with rank 2 perturbation

“…(ii) Let for any a < m, b > M , µ 0 = µ 0 (a) = (I 1 (a)) −1 , λ 0 = λ 0 (b) = (−I 2 (b)) −1 the equalities ∆ (1) µ0 (a) = 0 and ∆ So, in this paper, it was shown that there are two eigenvalues lying, correspondingly, to the left and to the right of the essential spectrum for the Friedrichs model h µ,λ . The existence of the threshold eigenvalues and virtual levels (threshold energy resonances) for a generalized Friedrichs model have been studied in [19][20][21]. We recall also that the symmetric operators of the form S := A ⊗ I + I ⊗ T , where A is symmetric and T = T * is (in general) unbounded, is considered in [22] and a boundary triplet Π S for S * preserving the tensor structure is constructed.…”

Estimates for the Bounds of the Essential Spectrum of a 2 × 2 Operator Matrix

“…It is remarkable that, the results about the essential spectrum and the number of the eigenvalues of  µ were announced without proofs in [19], and this paper is devoted to the detailed discussions of the results related with the essential spectrum of  µ .…”

“…It is important that in [19] it was found the critical value µ 0 of the coupling constant µ, to establish the existence of infinitely many eigenvalues lying in both sides of essential spectrum of  µ0 and to obtain an asymptotics for the number of these eigenvalues. The latter assertion seems to be quite new for the discrete models and similar result have not been obtained yet for the three-particle discrete Schrödinger operators and operator matrices in Fock space.…”