Two-sided estimates for the eigenvalues of the Laplace operator with Dirichlet boundary conditions and their application to problems in the mathematical theory of waveguides

Reliable two-sided algorithmically simple estimates for the eigenvalues of the Laplace operator in convex polygons with Dirichlet boundary conditions are obtained. These estimates are applied to the existence problem for the trapping modes of waveguides and to the problem of finding the frequency ranges for which the wave radiation occurs without resonance.