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

Abstract

We consider a 2 × 2 operator matrix Aμ, μ > 0, related with the lattice systems describing three particles in interaction, without conservation of the number of particles on a d-dimensional lattice. We obtain an analogue of the Faddeev type integral equation for the eigenfunctions of Aμ. We describe the two- and three-particle branches of the essential spectrum of Aμ via the spectrum of a family of generalized Friedrichs models. It is shown that the essential spectrum of Aμ consists of the union of at most three bounded closed intervals. We estimate the lower and upper bounds of the essential spectrum of Aμ with respect to the dimension d ∈ N of the torus Td and the coupling constant μ > 0.

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

“…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]. In [14][15][16], using the spectral information about the Hamiltonian H with the rank-1 generated kernel, the number of eigenvalues located respectively in the gap, inside, and below the bottom of the essential spectrum of the operator matrices is studied.…”