Spectral bounds for $f$-Laplace-type operators with applications to Betti numbers on gradient Ricci shrinkers

We study the spectrum of the $f$-Laplacian on complete gradient Ricci shrinkers. Upper and lower bounds for the $k$-th eigenvalue are established in terms of the volume growth rate. Both bounds are shown to be sharp in the exponent. The method extends to $f$-Laplace-type operators on vector bundles; as an application we obtain explicit upper bounds for the Betti numbers.