ABSTRACT.A model operator H associated with the energy operator of a system describing three particles in interaction, without conservation of the number of particles, is considered. The precise location and structure of the essential spectrum of H is described. The existence of infinitely many eigenvalues below the bottom of the essential spectrum of H is proved for the case where an associated generalized Friedrichs model has a resonance at the bottom of its essential spectrum. An asymptotics for the number N (z) of eigenvalues below the bottom of the essential spectrum is also established. The finiteness of eigenvalues of H below the bottom of the essential spectrum is proved if the associated generalized Friedrichs model has an eigenvalue with energy at the bottom of its essential spectrum.

A lattice model of radiative decay (so-called spin-boson model) of a two level atom and at most two photons is considered. The location of the essential spectrum is described. For any coupling constant the finiteness of the number of eigenvalues below the bottom of its essential spectrum is proved. The results are obtained by considering a more general model H for which the lower bound of its essential spectrum is estimated. Conditions which guarantee the finiteness of the number of eigenvalues of H, below the bottom of its essential spectrum are found. It is shown that the discrete spectrum might be infinite if the parameter functions are chosen in a special form.

We consider a family of 2 × 2 operator matrices Aµ(k), k ∈ T 3 := (−π, π] 3 , µ > 0, acting in the direct sum of zero-and one-particle subspaces of a Fock space. It is associated with the Hamiltonian of a system consisting of at most two particles on a three-dimensional lattice Z 3 , interacting via annihilation and creation operators. We find a set Λ := {k (1) , ..., k (8) } ⊂ T 3 and a critical value of the coupling constant µ to establish necessary and sufficient conditions for either z = 0 = min k∈T 3 σess(Aµ(k)) ( or z = 27/2 = max k∈T 3 σess(Aµ(k)) is a threshold eigenvalue or a virtual level of Aµ(k (i) ) for some k (i) ∈ Λ.

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В статье изучается существенный спектр модельного решетчатого гамильтониана для системы с переменным числом частиц 0 n 2 в квазиимпульсном представлении. Свойства спектра описаны в терминах граничных значений функции комплексного аргумента, имеющей смысл ядра комплемента Шура H11 − z − H12(H22 − z) −1 H * 12. Библиография: 13 названий.