2017/09/30 by Colin Guillarmou, Andrew Hassell, Katya Krupchyk
Mathematics · #Advanced Harmonic Analysis Research #Conic section #Dimension (graph theory) #Eigenvalues and eigenvectors #Euclidean geometry #Metric (unit) #Numerical methods in inverse problems #Operator (biology) #Spectral Theory in Mathematical Physics #Type (biology) #math.AP #math.SP #msc:35P15 #msc:42B37 #msc:58J40 #msc:58J50
paper · pdf · doi:10.2140/apde.2020.13.1633
published as Analysis & PDE 13 (2020) 1633-1670
openalex created_date 2017/10/06 · arxiv created 2019/09/23 · openalex publication_date 2020/09/12 · arxiv updated 2020/09/16 · openalex updated_date 2026/08/05
We study eigenvalues of non-self-adjoint Schrödinger operators on nontrapping asymptotically conic manifolds of dimension [math] . 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 [math] -norm of the potential, while the second one is a Lieb–Thirring-type bound controlling sums of powers of eigenvalues in terms of the [math] -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 [math] on [math] with [math] being a nontrapping compactly supported (or suitably short-range) perturbation of the Euclidean metric and [math] complex-valued.