Bounds on Accumulation Rates of Eigenvalues on Manifolds with Degenerating Metrics

We consider a family of manifolds with a class of degenerating warped product metrics $g_{ϵ}=\rho (ϵ,t{)}^{2a}d{t}^2+\rho (ϵ,t{)}^{2b}d{s}_M^2$, with $M$ compact, $\rho$ homogeneous degree one, $a\le -1$ and $b>0$. We study the Laplace operator acting on $L^2$ differential $p$-forms and give sharp accumulation rates for eigenvalues near the bottom of the essential spectrum of the limit manifold with metric $g_0$.