The heat equation and independence of the spectrum of the Hodge Laplacian on $\ell^p$

Abstract

We study the heat equation associated to the Hodge Laplacian on simplicial complexes. Using recently developed techniques for magnetic Schrödinger operators, we prove Davies-Gaffney-Grigoryan type estimates for the kernel of the heat semigroup on $\ell^2,$ which we then use to extend the semigroup to $\ell^p$ for $p\in[1,\infty]$ under suitable curvature and volume growth conditions. Furthermore, we establish $p$-independence of the Hodge Laplacian spectrum under the assumption of form bounded curvature and uniform subexponential volume growth. While the main focus of the paper is the Hodge Laplacian on simplicial complexes, the results are indeed proven for general positive magnetic Schrödinger operators on graphs.

Theorems & Definitions (60)

• Theorem 1.1: Heat equation
• Theorem 1.2: $p$-independence of the spectrum
• Proposition 2.1: Semigoup convergence
• Theorem 2.2: Davies-Gaffney-Grigoryan
• Lemma 2.3
• proof
• proof : Proof of Theorem \ref{['dgg']}
• Corollary 2.4
• proof
• Theorem 3.1: Form bounded potentials