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

Abstract

We study eigenvalues of non-self-adjoint Schrödinger 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 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-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.

References in corpus (3)

• Lieb-Thirring inequalities for Schrödinger operators with complex-valued potentials
• Schrödinger operator with non-zero accumulation points of complex eigenvalues
• Estimates for eigenvalues of Schrödinger operators with complex-valued potentials

Cited by in corpus (4)

• Counterexample to the Laptev--Safronov conjecture
• The Sobolev Inequalities on Real Hyperbolic Spaces and Eigenvalue Bounds for Schrödinger Operators with Complex Potentials
• Semiclassical limit of orthonormal Strichartz estimates on scattering manifolds
• Random Schrödinger operators with complex decaying potentials