Abstract : The first results presented here concern asymptotic methods for the study of the "continuity of the spectrum" of the Laplace operator acting on functions or differential forms on a compact manifold:
-excision of small tubular neighbourhood (with different boundary conditions)
-add of small handles
we obtain also asymptotics of the eigenforms and applications to the continuous spectrum of periodic manifolds.
The second part is concerned with pseudodifferential operators and semiclassical calculus:
-comparison between Dirichlet and Neumann spectrum for the Elasticity operator
-semiclassical localization of the joint spectrum of several commuting pseudodifferential operators.