The spectrum of the Laplacian on forms over flat manifolds

TL;DR

This paper proves that the spectrum of the Laplacian on k-forms over noncompact flat manifolds is always a connected closed interval, using a detailed structural decomposition of these manifolds.

Contribution

It establishes a complete characterization of the Laplacian spectrum on k-forms for noncompact flat manifolds, which was previously unknown.

Findings

Spectrum is always a connected closed interval

Spectrum starts at zero and extends continuously to a finite upper bound

Structural decomposition aids in spectral analysis

Abstract

In this article we prove that the spectrum of the Laplacian on $k$-forms over a noncompact flat manifold is always a connected closed interval of the nonnegative real line. The proof is based on a detailed decomposition of the structure of flat manifolds.