On the lower bound and upper bound of the Sum of Eigenvalues of the Fractional-Logarithmic Laplacian

TL;DR

This paper derives bounds for the sum of eigenvalues of the Fractional-Logarithmic Laplacian, addressing challenges posed by its non-monotonic Fourier symbol at low frequencies.

Contribution

It provides the first known bounds for eigenvalues of the Fractional-Logarithmic Laplacian, overcoming difficulties from its complex Fourier symbol behavior.

Findings

Established explicit lower and upper bounds for eigenvalues

Addressed the non-monotonic Fourier symbol challenge

Contributed to spectral theory of fractional operators

Abstract

We establish a lower bound and an upper bound to the sum of the Fractional-Logarithmic Laplacian. A main challenge in such a study comes from the fact that this operator has a Fourier symbol that is not globally monotone in its radial variable due to its low-frequency behavior.