Fundamental Gaps of the Fractional Schr\"odinger Operator

Abstract

We study asymptotically and numerically the fundamental gap -- the difference between the first two smallest (and distinct) eigenvalues -- of the fractional Schr\"{o}dinger 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 eigenfunctions 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 Schr\"{o}dinger operator. Extensions of these results for the FSO in the whole space and on bounded domains with periodic boundary conditions are presented.