Strict monotonicity and unique continuation for general non-local eigenvalue problems

TL;DR

This paper explores the relationship between the monotonicity of eigenvalues and the unique continuation property of eigenfunctions for non-local operators, including fractional Laplacians, revealing new theoretical insights.

Contribution

It establishes the equivalence between eigenvalue monotonicity and unique continuation for general non-local pseudo-differential operators, extending known results to broader classes.

Findings

Monotonicity of eigenvalues implies unique continuation of eigenfunctions.

Unique continuation results are obtained for fractional Laplacians.

The paper characterizes conditions under which eigenvalue monotonicity and unique continuation are equivalent.

Abstract

We consider the weighted eigenvalue problem for a general non-local pseudo-differential operator, depending on a bounded weight function. For such problem, we prove that strict (decreasing) monotonicity of the eigenvalues with respect to the weight function is equivalent to the unique continuation property of eigenfunctions. In addition, we discuss some unique continuation results for the special case of the fractional Laplacian.