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

In the present paper we consider a lattice spin-boson model with at most one photon A , which has a 2×2 block operator matrix representation. The essential spectrum of A is analyzed. We prove that the operator matrix A has four eigenvalues. We consider the case where the special integral is an infinite. The existence condition of the eigenvalues lying in and out of the essential spectrum are found. The results presented in this paper plays an important role when we study the location of the two-particle and three-particle branches of the essential spectrum of the lattice spin-boson Hamiltonian with at most two photons, and also to showing the finiteness of the number of its eigenvalues.

On the eigenvalues of the lattice spin-boson model with at most one photon

Dilmurodov,

Bahronov,

Khayitova

et al. 2024

2×2 operator matrix with real parameter and its spectrum

Dilmurodov,

Tosheva,

Turayeva

et al. 2024

In the present paper we consider a linear bounded self-adjoint 2×2 block operator matrix Aμ (so called generalized Friedrichs model) with real parameter μ ∈ R. It is associated with the Hamiltonian of a system consisting of at most two particles on a d -dimensional lattice Zd, interacting via creation and annihilation operators. Aμ is linear bounded self-adjoint operator acting in the two-particle cut subspace of the Fock space, that is, in the direct sum of zero-particle and one-particle subspaces of a Fock space. We find the essential and discrete spectra of the block operator matrix Aμ. The Fredholm determinant and resolvent operator associated to Aμ are constructed. The spectrum of Aμ plays an important role in the study of the spectral properties of the Hamiltonians associated with the energy operator of a lattice system describing two identical bosons and one particle, another nature in interactions, without conservation of the number of particles on a lattice.

Faddeev equation and its symmetric version for a three-particle lattice hamiltonian

Umirkulova,

Bahronov,

Tosheva

et al. 2024

In the present paper we consider the three-particle lattice Hamiltonian associated to a system of three particles on the d-dimensional lattice, where the role of two-particle discrete Schroedinger operators is played by a family of Friedrichs models. We define two bounded and self-adjoint so-called channel operators and prove that the essential spectrum of considered Hamiltonian is the union of spectra of the channel operators. Since the channel operators have a more simple structure than considered Hamiltonian, this fact plays an important role in the subsequent investigations of the essential spectrum. The spectrum of the constructed channel operators are described by the spectrum of the corresponding Friedrichs model. The Faddeev equation and its symmetric version for the eigenfunctions of the considered Hamiltonian are constructed.