Eigenvalue bounds for non-self-adjoint schrödinger operators with nontrapping metrics

We study eigenvalues of non-self-adjoint Schrödinger operators on nontrapping asymptotically conic manifolds of dimension n ≥ 3. Specifically, we are concerned with the following two types of estimates. The first one deals with Keller-type bounds on individual eigenvalues of the Schrödinger operator with a complex potential in terms of the L^p-norm of the potential, while the second one is a Lieb-Thirring-type bound controlling sums of powers of eigenvalues in terms of the L^p-norm of the potential. We extend the results of Frank (2011), Frank and Sabin (2017), and Frank and Simon (2017) on the Keller- and Lieb-Thirring-type bounds from the case of Euclidean spaces to that of nontrapping asymptotically conic manifolds. In particular, our results are valid for the operator δg + V on Rⁿ with g being a nontrapping compactly supported (or suitably short-range) perturbation of the Euclidean metric and V ∈ L^p complex-valued.