Various asymptotic expansions are derived for eigenvalues in the discrete spectrum of the boundary-value problem for the Laplace operator in the unit strip with the Dirichlet condition on its lateral sides everywhere with exception of an interval of length \(2\ell >0\) , where the Neumann condition is imposed (a planar quantum waveguide with “window”). Since the total multiplicity of the discrete spectrum grows indefinitely as \(\ell \rightarrow +\infty \) , there exists a sequence of critical lengths \(\{\ell ^*_m\}\) , for which the operator of the problem enjoys the threshold resonance. This phenomenon is characterized by the existence of a nontrivial bounded solution, that is, either trapped or almost standing wave, and provides miscellaneous near-threshold spectral anomalies. The quality of the threshold resonances is examined and asymptotic formulas for the values \(\ell ^*_m\) are obtained for large m. The analysis is systematically performed by methods of fracture mechanics. Bibliography: 58 titles.
