  A rigorous solution for the spectrum of quasioptical cylindrical cavity
resonator with a randomly rough side boundary has been obtained for the first
time. To accomplish this task, we have developed a novel method for variables
separation in wave equation, which enables one, in principle, to rigorously
examine any limiting case --- from negligibly weak to arbitrarily strong
disorder. It is shown that the effect of disorder-induced scattering can be
properly described in terms of two geometric potentials, specifically, the
"amplitude" and the "gradient" potentials, which appear in wave equation in the
course of conformal smoothing of the resonator boundaries. The scattering
resulting from the gradient potential appears to be dominant, and its impact on
the whole spectrum is governed by the unique sharpness parameter $\Xi$, the
mean tangent of the asperity slope. As opposed to the resonator with bulk
disorder, the distribution of nearest-neighbor spacings (NNS) in the
rough-resonator spectrum acquires Wigner-like features only when the wave
operator loses its unitarity, i.e., with the availability in the system of
either openness or dissipation channels. Our numeric experiments suggest that
in the absence of dissipation loss the random-rough resonator spectrum is
always regular, whatever the degree of roughness. Yet, the spectrum structure
is quite different in the domains of small and large values of the parameter
$\Xi$. For the dissipation-free resonator, the NNS distribution changes its
form with growing the asperity sharpness from Poissonian-like distribution in
the limit of $\Xi<<1$ to the bell-shaped distribution in the domain where
$\Xi>>1$.
