Communications in Mathematical Sciences

Volume 17 (2019)

Number 2

Fundamental gaps of the fractional Schrödinger operator

Pages: 447 – 471

DOI: https://dx.doi.org/10.4310/CMS.2019.v17.n2.a7

Authors

Weizhu Bao (Department of Mathematics, National University of Singapore)

Xinran Ruan (Department of Mathematics, National University of Singapore, Singapore; Sorbonne Université, CNRS, Université de Paris, Laboratoire J.-L. Lions, Paris, France)

Jie Shen (Department of Mathematics, Purdue University, West Lafayette, Indiana, U.S.A.)

Changtao Sheng (Fujian Provincial Key Laboratory on Mathematical Modeling & High Performance Scientific Computing, and School of Mathematical Sciences, Xiamen University, Xiamen, Fujian, China)

Abstract

We study asymptotically and numerically the fundamental gap – the difference between the first two smallest (and distinct) eigenvalues – of the fractional Schrödinger operator (FSO) and formulate a gap conjecture on the fundamental gap of the FSO. We begin with an introduction of the FSO on bounded domains with homogeneous Dirichlet boundary conditions, while the fractional Laplacian operator defined either via the local fractional Laplacian (i.e. via the eigenfunction decomposition of the Laplacian operator) or via the classical fractional Laplacian (i.e. zero extension of the eigenfunctions outside the bounded domains and then via the Fourier transform). For the FSO on bounded domains with either the local fractional Laplacian or the classical fractional Laplacian, we obtain the fundamental gap of the FSO analytically on simple geometry without potential and numerically on complicated geometries and/or with different convex potentials. Based on the asymptotic and extensive numerical results, a gap conjecture on the fundamental gap of the FSO is formulated. Surprisingly, for two and higher dimensions, the lower bound of the fundamental gap depends not only on the diameter of the domain, but also the diameter of the largest inscribed ball of the domain, which is completely different from the case of the Schrödinger operator. Extensions of these results for the FSO in the whole space and on bounded domains with periodic boundary conditions are presented.

Keywords

fractional Schrödinger operator, fundamental gap, gap conjecture, local fractional Laplacian, classical fractional Laplacian, homogeneous Dirichlet boundary condition, periodic boundary condition

2010 Mathematics Subject Classification

05B40, 26A33, 35J10, 35P15, 35R11

The research of W.B. was supported by the Academic Research Fund of Ministry of Education of Singapore grant R-146-000-223-112.

The work of J.S. was supported in part by NSF DMS-1620262, DMS-1720442 and DMS-1720442.

The work of J.S. was supported in part by NSF DMS-1620262, DMS-1720442 and DMS-1720442.

Received 20 January 2018

Received revised 7 December 2018

Accepted 7 December 2018

Published 8 July 2019