Stability estimates for the lowest eigenvalue of a Schr\"odinger operator
Analysis of PDEs
2013-05-15 v2 Mathematical Physics
math.MP
Spectral Theory
Abstract
There is a family of potentials that minimize the lowest eigenvalue of a Schr\"odinger eigenvalue under the constraint of a given L^p norm of the potential. We give effective estimates for the amount by which the eigenvalue increases when the potential is not one of these optimal potentials. Our results are analogous to those for the isoperimetric problem and the Sobolev inequality. We also prove a stability estimate for H\"older's inequality, which we believe to be new.
Cite
@article{arxiv.1301.5032,
title = {Stability estimates for the lowest eigenvalue of a Schr\"odinger operator},
author = {Eric A. Carlen and Rupert L. Frank and Elliott H. Lieb},
journal= {arXiv preprint arXiv:1301.5032},
year = {2013}
}
Comments
20 pages; reference added; minor change in the statement and proof of Theorem 3.1