Bound states of the Schrödinger operator of a system of three bosons on a lattice

We consider the Hamiltonian $H_\mu$ of a system of three identical quantum particles (bosons) moving on a $d$-dimensional lattice $\mathbb Z^d$, $d=1,2$, and coupled by an attractive pairwise contact potential $\mu<0$. We prove that the number of bound states of the corresponding Schrödinger operator $H_\mu(K)$, $K\in\mathbb T^d$, is finite and establish the location and structure of its essential spectrum. We show that the bound state decays exponentially at infinity and that the eigenvalue and the corresponding bound state as functions of the quasimomentum $K\in\mathbb T^d$ are regular.