Instantons in $N$-particle systems on a lattice

Tunnel asymptotics for the ground state wave function of a system of bosons with binary interaction is constructed. For various relations of parameters similar to the Bogolyubov relation in the theory of superfluidity, two possible cases are realized, i. е., either the ground state is space-uniform, or there exist two different asymptotic states with a symmetry between them. The latter may be interpreted as an analog of vortex flows in the theory of superfluidity. Using the Maslov quasiclassical methods, the exponentially small energy splitting of excited states is calculated.