Abstract：

We consider a non-compact Riemannian manifold $M$ having ends equipped with asymptotically warped product metric. We allow any topology and metric for the finite part of $M$. As for the volume growth of each end, we assume any polynomial or exponential order. Studying the spectral properties of the Laplacian of $M$, we show that the S-matrix determines the manifold $M$. We can also include polynomially or exponentially shrinking cusp ends. This is a joint work with Matti Lassas in the university of Helsinki.