The generalised P\'{o}lya conjecture for the Dirichlet eigenvalues

TL;DR

This paper proves the generalized Pólya conjecture for Dirichlet eigenvalues of the fractional Laplacian, establishing a lower bound for eigenvalues in bounded domains, extending classical spectral inequalities.

Contribution

It establishes the generalized Pólya conjecture for fractional Laplacian eigenvalues, a significant extension of classical spectral bounds to fractional operators.

Findings

Proves the inequality for all eigenvalues $k=1,2,3,...$

Extends Pólya's conjecture to fractional Laplacians with $eta eq 2$

Provides bounds depending on domain volume and eigenvalue index

Abstract

In this paper, we prove the Generalized P\'{o}lya conjecture for the Dirichlet eigenvalues. In other words, we show that $\lambda_k(\alpha) \ge \frac{(2\pi)^{\alpha} k^{\alpha/n}}{\big(\omega_n \cdot {vol}(\Omega)\big)^{\alpha/n}}, \quad\, {for}\;\; k=1,2,3,....$ where $\lambda_k(\alpha)$ is the $k$-th Dirichlet eigenvalue for the fractional Laplacian $(-\Delta)^{\alpha/2}$ with $\alpha\in (0,2]$ in a bounded domain $\Omega\subset {\Bbb R}^n$.