  We consider the quantum Lobachevsky space ${\bf L}_q^3$, which is defined as
subalgebra of the Hopf algebra ${\cal A}_q(SL_2({\bf C}))$. The Iwasawa
decomposition of ${\cal A}_q(SL_2({\bf C}))$ introduced by Podles and
Woronowicz allows to consider the quantum analog of the horospheric coordinates
on ${\bf L}_q^3$. The action of the Casimir element, which belongs to the dual
to ${\cal A}_q$ quantum group $U_q(SL_2({\bf C}))$, on some subspace in ${\bf
L}_q^3$ in these coordinates leads to a second order difference operator on the
infinite one-dimensional lattice. In the continuos limit $q\rightarrow 1$ it is
transformed into the Schr\"{o}dinger Hamiltonian, which describes zero modes
into the Liouville field theory (the Liouville quantum mechanics). We calculate
the spectrum (Brillouin zones) and the eigenfunctions of this operator. They
are $q$-continuos Hermit polynomials, which are particular case of the
Macdonald or Rogers-Askey-Ismail polynomials. The scattering in this problem
corresponds to the scattering of first two level dressed excitations in the
$Z_N$ Baxter model in the very peculiar limit when the anisotropy parameter
$\ga$ and $N~\rightarrow\infty$, or, equivalently, $(\ga, N)\rightarrow 0$.
