Energy-momentum spectrum of some two-particle lattice Schrödinger Hamiltonians

We determine the excitation spectrum of some one and two-particle Z(d) lattice Schrödinger Hamiltonians. They occur as approximate Hamiltonians for the low-lying energy-momentum spectrum of diverse infinite lattice nonlinear quantum systems. A unitary staggering transformation relates the low-energy-momentum spectrum to the high-energy-momentum spectrum of the transformed operators. A feature for the one-particle repulsive delta function Hamiltonian is that, in addition to the continuous band spectrum, there is a bound state above the band, and the repulsive case spectrum and scattering can be obtained from the attractive potential case by staggering. For the two-particle pair potential Hamiltonian, there are commuting self-adjoint energy-momentum operators, and we determine the joint spectrum. For the case of a lambda delta pair potential, and equal particle masses, for arbitrarily small /lambda/, lambda < 0, and d >or = 3, there is no bound state for small system momentum, but a bound state exists below the band if the momentum is large. We find that the binding energy is an increasing function of the system momentum. The existence of this bound state is in contrast with the continuum case, where the Birman-Schwinger bound excludes negative-energy bound states for small couplings; this bound state is absent if the two masses are different. Other spectral results are also obtained for the large coupling case. An eigenfunction expansion that uses products of plane waves in the sum and difference coordinates is used to obtain the spectral results.