Counterexample to the second eigenfunction having one zero for a non-local Schrodinger operator

Abstract

We demonstrate that the second eigenfunction of a perturbed fractional Laplace operator on a bounded interval can exhibit two sign changes, in stark contrast with the classical expectation that it should have exactly one zero. Our construction employs the Kato-Rellich regular perturbation theory to analyse an infinite potential well eigenvalue problem, and then uses a compactness and energy-minimisation argument to extend this counterexample to finite potential wells. Although our detailed analysis focuses on the case where s takes value 1/2 (the Cauchy process), our approach indicates that similar phenomena occur for other rational values of s in (0, 1). At the time of writing, this result provides one of the first rigorous insights into the qualitative behaviour of eigenfunctions for perturbed nonlocal Schrodinger operators.