Structure of the spectrum of a periodic family of identical cells connected by converging apertures
For \(d\geq 2\), the authors define \(\Pi\), the \(d\)-dimensional waveguide comprised by the union of identical cells (its elements belong to \(\Omega\), a bounded subset of an explicit strip in \(\mathbb R^{d}\)) and apertures (their sizes decay). The authors show that the spectra of \(DP_\Pi\) and \(DP_\Omega\), the Dirichlet problem on \(\Pi\) and on \(\Omega\), respectively, are closely related. Precisely, they show that the essential spectrum of \(DP_\Pi\) is the family of the eigenvalues of \(DP_\Omega\) and \(DP_\Pi\) has an increasing sequence of eigenvalues converging to the first eigenvalue of \(DP_\Omega\) (Theorem, section 2). Among the apparatus used on the existence of the essential spectrum, they apply the Weyl criteria, and in the rest of the proof, they use the Hardy inequality -- the logarithm Hardy inequality for the planar waveguide -- and the Friedrichs inequality on the semi-hypersphere of the hypersphere \(\mathbb S^{d-1}\).
scientific article; zbMATH DE number 6347702