Spectral Properties of Non-local Schr\"{o}dinger Operators: Techniques and a Counterexample

Abstract

This thesis aims to expand our understanding of the spectral properties of non-local Schr\"{o}dinger operators on an open set in Euclidean space. The first part concerns extending the method of modulus of continuity for solutions of parabolic equations---as used, for instance, to prove the Fundamental Gap Conjecture---to solutions of non-local heat equations on $\R^n$ and in dimension one with a non-local Neumann boundary condition. Specifically, we show that if a solution of a non-local heat equation has an initial modulus of continuity satisfying simple criteria, then this modulus of continuity is preserved at all subsequent times. In the process of trying to generalise our result in one dimension, we found a counterexample suggesting that a non-local analogue of the Payne-Weinberger inequality would depend on more than the diameter of a bounded (convex) domain.

In the second part, we construct a counterexample demonstrating that the second eigenfunction of a perturbed fractional Laplace operator on a bounded interval, with Dirichlet `boundary' data off the interval, can exhibit more than one sign change. This stands in stark contrast to 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 $s=1/2$ (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 non-local Schr\"{o}dinger operators.