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

TL;DR

This paper presents a counterexample showing that the second eigenfunction of a perturbed fractional Laplace operator can have two zeros, challenging classical assumptions and providing new insights into nonlocal Schrödinger operators.

Contribution

It introduces the first rigorous counterexample demonstrating multiple sign changes in the second eigenfunction of a nonlocal Schrödinger operator.

Findings

Second eigenfunction can have two zeros, not one.

Counterexample constructed for fractional Laplace operator with s=1/2.

Method suggests similar phenomena for other rational s in (0,1).

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 an 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 strongly suggests 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.