Number of bound states of the Schroedinger operator of a system of three bosons in an optical lattice

We consider the Hamiltonian $\hat {\mathrm{H}}_{\mu}$ of a system of three identical particles(bosons) on the $d-$ dimensional lattice $\Z^d, d=1,2$ interacting via pairwise zero-range attractive potential $\mu<0$. We describe precise location and structure of the essential spectrum of the Schr\"odinger operator $H_\mu(K),K\in \T^d$ associated to $\hat {\mathrm{H}}_\mu$ and prove the finiteness of the number of bound states of $H_\mu(K),K\in \T^d$ lying below the bottom of the essential spectrum. Moreover, we show that bound states decay exponentially at infinity and eigenvalues and corresponding bound states of $H_\mu(K),K\in \T^d$ are regular as a function of center of mass quasi-momentum $K\in \T^d$.