THE SPECTRUM OF THE HODGE LAPLACIAN FOR A DEGENERATING FAMILY OF HYPERBOLIC 3-MANIFOLDS

We consider a sequence (M(n))(n=1)(infinity) of compact hyperbolic man ifolds converging to a complete hyperbolic manifold M(0) with cusps. T he Laplace operator acting on the space of L(2) differential forms on M(0) has continuous spectrum filling the half-line [0, infinity). One expects therefore that the spectra of this operator on M(n) accumulate to produce the continuous spectrum of the limiting manifold. We prove that this is the case and obtain a sharp estimate of the rate of accu mulation.