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

Abstract

We study eigenvalues of non-self-adjoint Schrödinger operators on nontrapping asymptotically conic manifolds of dimension $n\ge 3$. 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 ${\mathrm{\Delta }}_g+V$ on ${ℝ}^n$ with $g$ being a nontrapping compactly supported (or suitably short-range) perturbation of the Euclidean metric and $V\in L^p$ complex-valued.