The Essential Spectrum of a Three Particle Schrödinger Operator on Lattices
Abstract
We consider the Hamiltonian $$\textrm{H}_{\mu\lambda},\mu,\lambda\in\mathbb{R}$$ of a system of three-particles (two identical bosons and one different particle) moving on the lattice $${\mathbb{Z}}^{d},\,d=1,2$$ interacting through zero-range pairwise potentials $$\mu\neq 0$$ and $$\lambda\neq 0$$ . The essential spectrum of the three-particle discrete Schrödinger operator $$H_{\mu\lambda}(K),\,K\in\mathbb{T}^{d}$$ , being the three-particle quasi-momentum, is described by means of the spectrum of non-perturbed three-particle operator $$H_{0}(K)$$ and the two-particle discrete Schrödinger operator $$h_{\mu}(k),h_{\lambda,\gamma}(k),k\in\mathbb{T}^{d},\gamma>0$$ . It is established that the essential spectrum of the three-particle discrete Schrödinger operator $$H_{\mu\lambda}(K),\,K\in\mathbb{T}^{d}$$ consists of no more than three bounded closed intervals.