On the eigenfunctions associated with the high frequencies in systems witha concentrated mass

We consider the vibrations of a system consisting of the domain Omega of R- N, N = 2, 3, that contains a small region with diameter depending on a smal l parameter epsilon. The density is of order O(epsilon(-m)) in the small re gion, the concentrated mass, and it is O(1) outside; m is a parameter, m gr eater than or equal to 2. We study the asymptotic behaviour, as epsilon --> 0, of the eigenvalues of order O(1), the high frequencies when m > 2, and the corresponding eigenfunctions of the associated spectral problem. We pro vide information on the structure of these eigenfunctions. We also check th eoretical results with explicit calculations for the dimensions N = 1 and N = 2 and give correcting terms for the eigenfunctions. (C) Elsevier, Paris.