Eigenvalue bounds for non-self-adjoint Schr\"odinger operators with non-trapping metrics

TL;DR

This paper extends eigenvalue bounds for non-self-adjoint Schr"odinger operators from Euclidean spaces to non-trapping asymptotically conic manifolds, providing Keller and Lieb-Thirring type estimates in this broader geometric setting.

Contribution

It generalizes Keller and Lieb-Thirring eigenvalue bounds to non-trapping asymptotically conic manifolds, including Euclidean spaces with perturbations.

Findings

Established Keller type bounds for eigenvalues in terms of $L^p$-norm of potential.

Derived Lieb-Thirring type bounds controlling sums of eigenvalues.

Extended previous Euclidean results to non-trapping asymptotically conic manifolds.

Abstract

We study eigenvalues of non-self-adjoint Schr\"odinger operators on non-trapping 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 Schr\"odinger 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-Sabin (2017), and Frank-Simon (2017) on the Keller and Lieb-Thirring type bounds from the case of Euclidean spaces to that of non-trapping asymptotically conic manifolds. In particular, our results are valid for the operator $\Delta_g+V$ on $\mathbb{R}^n$ with $g$ being a non-trapping compactly supported (or suitably short range) perturbation of the Euclidean metric and $V\in L^p$ complex valued.