Ahmad M. Khalkhuzhaev

arXiv2015

The existence of an isolated band in a system of three particles in an optical lattice

We prove the existence of two-and three-particle bound states of the Schrödinger operators $h_μ(k),k\in \T^d$ and $H_μ(K),K\in \T^d$ associated to Hamiltonians $\mathrm{h}_μ$ and $\mathrm{H}_μ$ of a system of two and three identical bosons on the lattice $\Z^d, d=1,2$ interacting via pairwise zero-range attractive $μ<0$ or repulsive $μ>0$ potentials. As a consequence we show the existence of an isolated band in the two and three bosonic systems in an optical lattice.

math.SP↗

arXiv2015

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

We consider the Hamiltonian $\hat {\mathrm{H}}_μ$ 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 $μ<0$. We describe precise location and structure of the essential spectrum of the Schrödinger operator $H_μ(K),K\in \T^d$ associated to $\hat {\mathrm{H}}_μ$ and prove the finiteness of the number of bound states of $H_μ(K),K\in \T^d$ 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 $H_μ(K),K\in \T^d$ are regular as a function of center of mass quasi-momentum $K\in \T^d$.

math.SP↗