Kroger's upper bound types for the Dirichlet eigenvalues of the fractional Laplacian

TL;DR

This paper derives an upper bound for the sum of Dirichlet eigenvalues of the fractional Laplacian, advancing understanding of spectral properties of non-local operators through precise Rayleigh quotient calculations.

Contribution

It introduces a new upper bound for eigenvalues of the fractional Laplacian's Dirichlet problem using refined Rayleigh quotient techniques.

Findings

Established an explicit upper bound for eigenvalue sums.

Provided a novel computational approach for Rayleigh quotients.

Enhanced spectral theory for non-local operators.

Abstract

We establish an upper bound of the sum of the eigenvalues for the Dirichlet problem of the fractional Laplacian. Our result is obtained by a subtle computation of the Rayleigh quotient for specific functions.