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.
- Publication:
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arXiv e-prints
- Pub Date:
- September 2017
- DOI:
- 10.48550/arXiv.1709.09759
- arXiv:
- arXiv:1709.09759
- Bibcode:
- 2017arXiv170909759G
- Keywords:
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- Mathematics - Analysis of PDEs;
- Mathematics - Spectral Theory;
- 35P15;
- 42B37;
- 58J50;
- 58J40
- E-Print:
- Analysis &