We prove a uniform lower bound for the difference λ₂ - λ₁ between the first two eigenvalues of the fractional Schrödinger operator , α ∈ (1,2), with a symmetric single-well potential V in a bounded interval (a,b), which is related to the Feynman-Kac semigroup of the symmetric α-stable process killed upon leaving (a,b). “Uniform” means that the positive constant appearing in our estimate is independent of the potential V. In the general case of α ∈ (0,2), we also find a uniform lower bound for the difference λ⁎ - λ₁, where λ⁎ denotes the smallest eigenvalue corresponding to an antisymmetric eigenfunction. One of our key arguments used in proving the spectral gap lower bound is a certain integral inequality which is known to be a consequence of the Garsia-Rodemich-Rumsey lemma. We also study some basic properties of the corresponding eigenfunctions.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm209-3-5,
author = {Kamil Kaleta},
title = {Spectral gap lower bound for the one-dimensional fractional Schr\"odinger operator in the interval},
journal = {Studia Mathematica},
volume = {209},
year = {2012},
pages = {267-287},
zbl = {1257.47049},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm209-3-5}
}
Kamil Kaleta. Spectral gap lower bound for the one-dimensional fractional Schrödinger operator in the interval. Studia Mathematica, Tome 209 (2012) pp. 267-287. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm209-3-5/