Some Subtleties in the Relationships among Heat Kernel Invariants, Eigenvalue Distributions, and Quantum Vacuum Energy

TL;DR

This paper clarifies subtle aspects of eigenvalue density asymptotics and their relation to heat kernel invariants and quantum vacuum energy, addressing recent observations within established spectral distribution frameworks.

Contribution

It elucidates the connections among spectral invariants, eigenvalue distributions, and quantum vacuum energy, clarifying recent observations in the context of spectral asymptotics.

Findings

Clarification of the relationship between heat kernel invariants and eigenvalue distributions.

Integration of recent observations into the established spectral asymptotics framework.

Improved understanding of the asymptotic behavior of Green functions in Casimir physics.

Abstract

A common tool in Casimir physics (and many other areas) is the asymptotic (high-frequency) expansion of eigenvalue densities, employed as either input or output of calculations of the asymptotic behavior of various Green functions. Here we clarify some fine points and potentially confusing aspects of the subject. In particular, we show how recent observations of Kolomeisky et al. [Phys. Rev. A 87 (2013) 042519] fit into the established framework of the distributional asymptotics of spectral functions.