Analytic Properties of Spectral Functions

Abstract

In the previous chapter we have witnessed the interplay between the spectral zeta functions and the associated heat traces (cf. Proposition 2.10). We have also learned, in Sect. 2.2.2, how to exploit the Laplace transform to compute the spectral action from a given heat trace. In this chapter we further explore the connections between the spectral functions unravelling the intimate relationship between the meromorphic continuation of a zeta function and the asymptotic expansion of the corresponding heat trace. We utilise the latter to establish the sought asymptotic expansion of the spectral action at large energies. Finally, we ponder the possibility of obtaining convergent, rather than only asymptotic, formulae for this action.