We prove weighted uniform estimates for the resolvent of the Laplace operator in Schatten spaces, on non-trapping asymptotically conic manifolds of dimension $n\ge 3$, generalizing a result of Frank and Sabin, obtained in the Euclidean setting. As an application of these estimates we establish Lieb-Thirring type bounds for eigenvalues of Schr\"odinger operators with complex potentials on non-trapping asymptotically conic manifolds, extending those of Frank, Frank and Sabin, and Frank and Simon proven in the Euclidean setting. In particular, our results are valid for the metric Schr\"odinger operator in the Euclidean space, with a metric being a sufficiently small compactly supported perturbation of the Euclidean one. To the best of our knowledge, these are the first Lieb-Thirring type bounds for non-self-adjoint elliptic operators, with principal part having variable coefficients. 查看全文>>