Non-perturbative results for the spectrum of surface-disordered waveguides

We calculated the spectrum of normal scalar waves in a planar waveguide with absolutely soft randomly rough boundaries beyond the perturbation theories in the roughness heights and slopes, basing on the exact boundary scattering potential. The spectrum is proved to be a nearly real non-analytic function of the dispersion ${\zeta }^2$ of the roughness heights (with square-root singularity) as ${\zeta }^2\to 0$. The opposite case of large boundary defects is summarized.