On Different Quasimodes for the Homogenization of Steklov-Type Eigenvalue Problems

Abstract

AbstractRoughly speaking, a quasimode for an operator with a discrete spectrum on a Hilbert space can be defined as a pair (\(\tilde{w}, \mu\)), where \(\tilde{w}\) is a function approaching a certain linear combination of eigenfunctions associated with the eigenvalues of the operator in a “small interval” [\(\mu - r, \mu + r\)]. The remainder r also deals with the discrepancies between \(\tilde{w}\) and the eigenfunctions.The value of the quasimodes in describing asymptotics for low and high frequency vibrations in certain singularly perturbed spectral problems, which depend on a small parameter ε, has been made clear recently in many papers. We refer to [Pe08] for an abstract general framework that can be applied to several problems of spectral perturbation theory and to [LoPe03] and [SaSa89] for a large variety of these problems. As a matter of fact, for these problems, the spaces and the operators under consideration depend on the parameter of perturbation, and the function \(\tilde{w}\) and the numbers μ and r arising in the definition of a quasimode can also depend on this parameter.KeywordsHarmonic FunctionLocal ProblemDiscrete SpectrumSpectral ProblemHigh Frequency VibrationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.