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

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 analytic perturbation theory of Kato and Rellich for degenerate eigenvalues 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 s = 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 Schrödinger operators.