Fundamental Gaps of the Fractional Schr\"odinger Operator

TL;DR

This paper investigates the fundamental gap of the fractional Schrödinger operator on various domains, providing analytical and numerical results, and proposes a new gap conjecture highlighting the influence of domain geometry.

Contribution

It introduces a gap conjecture for the fractional Schrödinger operator, supported by analytical and numerical analysis on different geometries and boundary conditions.

Findings

The fundamental gap depends on the domain's diameter and the largest inscribed ball's diameter.

Analytical solutions are obtained for simple geometries without potential.

Numerical results extend to complex geometries and various potentials.

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.