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We calculated the spectrum of normal scalar waves in a planar waveguide with absolutely soft randomly rough boundaries. Our approach is beyond perturbation theories in the roughness heights and slopes and is based instead on the exact boundary scattering potential. The spectrum is proved to be a nearly real nonanalytic function of the dispersion zeta(2) of the roughness heights (with square-root singularity) as zeta(2)?0 . The opposite case of large boundary defects is summarized.

Nonperturbative results for the spectrum of surface-disordered waveguides