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

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