On the Existence of Eigenvalues of the Three-Particle Discrete Schrödinger Operator

We consider the three-particle Schr odinger operator H-mu,H-lambda,H-gamma(K),K is an element of T-3, associated with a system of three particles (of which two are bosons with mass1and one is arbitrary with mass m=1/gamma <1) coupled by pairwise contact potentials mu > 0 and lambda > 0 on the three-dimensional lattice Z(3). We prove that there exist critical mass ratio values gamma= gamma(1)and gamma =gamma(2) such that for sufficiently large mu>0 and fixed lambda> 0 the operator H-mu,H-lambda,H-gamma (0),0=(0,0,0), has at least one eigenvalue lying to the left of the essential spectrum for gamma is an element of(0,gamma(1)), at least two such eigenvalues for gamma is an element of(gamma(1),gamma(2)), and at least four such eigenvalues for gamma is an element of(gamma(2),+infinity)