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

## Abstract

Roughly 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.

## Keywords

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