On the spectrum of the Laplace-Beltrami operator for $p$-forms for a class of warped product metrics

We explicitely compute the essential spectrum of the Laplace-Beltrami operator for $p$-forms for the class of warped product metrics $d\sigma^2= y^{2a}dy^2 + y^{2b}d\theta_{\partial M}^2$,where$y$ is a boundary defining function on a compact manifold with boundary $M$.