Spectrum of the Laplacian on radial graphs

TL;DR

This paper proves that the spectrum of the Laplace operator on complete radial graphs in Euclidean space is the entire non-negative real line, providing a spectral characterization of such hypersurfaces.

Contribution

It establishes that the spectrum of the Laplacian on complete radial graphs in Euclidean space is exactly [0, ∞), a new spectral property for these hypersurfaces.

Findings

Spectrum of Laplacian on radial graphs is [0, ∞)

Complete radial graphs have continuous non-negative spectrum

Spectral characterization aids understanding of geometric analysis on hypersurfaces

Abstract

We prove that if $M$ is a complete hypersurface in $\mathbb{R}^{n+1}$ which is graph of a real radial function, then the spectrum of the Laplace operator on M is the interval $[0,\infty)$.