Contents Online
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
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