The spectrum of the perturbed Laplace operator on the fundamental domain of a discrete group on the Lobachevskii plane

Let $C(z)$ be a real valued function decreasing in some sense in the neighbourhoods of the cusps of the fundamental domains and
$$ Lu=-y^2\Delta u+C(z)u. $$
Then $\sigma_{\mathrm{ac}}(L)=[1/4,+\infty)$; $\sigma_{\mathrm{sing}}(L)=\varnothing$; $\sigma_{\mathrm{pp}}(L)$ – is a discrete set of eigenvalues of finite multiplicities.