Spectral gap lower bound for the one-dimensional fractional Schrödinger operator in the interval

Kamil Kaleta¹

¹ Institute of Mathematics and Computer Science Wrocław University of Technology Wybrzeże Wyspiańskiego 27 50-370 Wrocław, Poland

Studia Mathematica, Tome 209 (2012) no. 3, pp. 267-287

Résumé

We prove a uniform lower bound for the difference $\lambda _2 - \lambda _1$ between the first two eigenvalues of the fractional Schrödinger operator $(-{\mit\Delta} )^{\alpha /2} + V$, $\alpha \in (1,2)$, with a symmetric single-well potential $V$ in a bounded interval $(a,b)$, which is related to the Feynman–Kac semigroup of the symmetric $\alpha $-stable process killed upon leaving $(a,b)$. “Uniform” means that the positive constant $C_{\alpha }$ appearing in our estimate $\lambda _2 - \lambda _1 \geq C_{\alpha } (b-a)^{-\alpha }$ is independent of the potential $V$. In the general case of $\alpha \in (0,2)$, we also find a uniform lower bound for the difference $\lambda _{*} - \lambda _1$, where $\lambda _{*}$ 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.

Détail

DOI : 10.4064/sm209-3-5

Keywords: prove uniform lower bound difference lambda lambda between first eigenvalues fractional schr dinger operator mit delta alpha alpha symmetric single well potential bounded interval which related feynman kac semigroup symmetric alpha stable process killed leaving uniform means positive constant alpha appearing estimate lambda lambda geq alpha b a alpha independent potential nbsp general alpha uniform lower bound difference lambda * lambda where lambda * denotes smallest eigenvalue corresponding antisymmetric eigenfunction key arguments proving spectral gap lower bound certain integral inequality which known consequence garsia rodemich rumsey lemma study basic properties corresponding eigenfunctions