L^p-spectral theory for the Laplacian on forms

2024/01/04 by Nelia Charalambous, Charalambous, Nelia, Zhiqin Lu +1
Mathematics · Computer Science · #Spectral Theory in Mathematical Physics #Advanced Mathematical Modeling in Engineering #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2401.02136

Abstract

In this article, we find sufficient conditions on an open Riemannian manifold so that a Weyl criterion holds for the L^p-spectrum of the Laplacian on k-forms, and also prove the decomposition of the L^p-spectrum depending on the order of the forms. We then show that the resolvent set of an operator such as the Laplacian on L^p lies outside a parabola whenever the volume of the manifold has an exponential volume growth rate, removing the requirement on the manifold to be of bounded geometry. We conclude by providing a detailed description of the L^p spectrum of the Laplacian on k-forms over hyperbolic space.

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