Ansatz of Hans Bethe for a two-dimensional lattice Bose gas

The method of q-oscillator lattices, proposed recently in Bazhanov and Sergeev 2005 (Preprint hep-th/0509181), provides the tool for a construction of various integrable models of quantum mechanics in (2 + 1)-dimensional spacetime. In contrast to any one-dimensional quantum chain, its two-dimensional generalizations - quantum lattices - admit different geometrical structures. In this paper, we consider the q-oscillator model on a special lattice. The model may be interpreted as a two-dimensional lattice Bose gas. The most remarkable feature of the model is that it allows the coordinate Bethe ansatz: the p-particles' wavefunction is the sum of plane waves. Consistency conditions is the set of 2p equations for p one-particle wave vectors. These 'Bethe ansatz' equations are the main result of this paper.