LIOUVILLE QUANTUM-MECHANICS ON A LATTICE FROM GEOMETRY OF QUANTUM LORENTZ GROUP

We consider the quantum Lobachevsky space L(q)3, which is defined as a subalgebra of the Hopf algebra A(q)(SL2(C)). The Iwasawa decompositio n of A(q)(SL2(C)) introduced by Podles and Woronowicz allows us to con sider the quantum analogue of the horospheric coordinates on L(q)3. Th e action of the Casimir element, which belongs to the dual to A(q) qua ntum group U(q)(SL2(C)), on some subspace in L(q)3 in these coordinate s leads to a second-order difference operator on the infinite one-dime nsional lattice. In the continuous limit q --> 1 it is transformed int o the Schrodinger Hamiltonian, which describes zero modes into the Lio uville field theory (the Liouville quantum mechanics). We calculate th e spectrum (Brillouin zones) and the eigenfunctions of this operator. They are q-continuous Hermite polynomials, which are particular cases of the Macdonald or Rogers-Askey-Ismail polynomials. The scattering in this problem corresponds to the scattering of the first two-level dre ssed excitations in the Z(N) model in the very peculiar limit when the anisotropy parameter gamma and N --> infinity, or, equivalently, (gam ma, N) --> 0.