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

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.