Higher-dimensional fractional-Laplacian eigenvalue asymptotics without non-periodicity

Remove the standard non-periodicity condition from the two-term Weyl law for the unsmoothed eigenvalue counting function of the fractional Laplacian on bounded higher-dimensional domains, and derive precise asymptotics for individual eigenvalues in such domains.

Background

The paper establishes refined eigenvalue asymptotics for the zero-exterior fractional Laplacian on a bounded one-dimensional interval. In the discussion of higher-dimensional extensions, it notes that one-term Weyl laws and several two-term results are already available for bounded domains, including a two-term Weyl law for the unsmoothed eigenvalue counting function under a standard non-periodicity condition.

The authors explicitly identify two unresolved extensions: eliminating that non-periodicity hypothesis and obtaining asymptotics for individual eigenvalues, rather than only for the counting function or eigenvalue sums. These problems concern the higher-dimensional fractional Laplacian on bounded domains in dimensions at least two.