Remark on Justification of Asymptotics of Spectra of Cylindrical Waveguides with Periodic Singular Perturbations of Boundary and Coefficients

To perform an asymptotic analysis of spectra singularly perturbed periodic waveguides, it is required to estimate remainders expansions eigenvalues a model problem on the periodicity cell uniformly with respect Floquet parameter. We propose two approaches this problem. The first based max–min principle and sufficiently easily realized, but has restricted application area. second more universal, technically complex since prove unique solvability for some value spectral parameter in nonempty closed segment, which verified by constructing almost inverse operator inhomogeneous variational setting. consider boundary problems simplest cell: rectangle row fine holes.