Jonathan Stanfill
The Ohio State University
Title
On applications and the analysis of spectral zeta functions
Abstract
This talk will concern the analysis of spectral zeta functions which represent generalizations of the Riemann zeta function, replacing integers with eigenvalues of a differential operator. The analytic continuation of the zeta function can then be related to many properties of interest. This ranges from geometric information of manifolds via the small-time asymptotic behavior of the trace of the heat kernel to characteristics of quantum fields via the zeta regularized functional determinant. While such regularization methods can be traced back to Hardy and Littlewood, and include Stephen Hawking as one of its original investigators applying them in physical problems, these methods continue to prove useful in investigating the Casimir effect, the cosmological constant, and Bose-Einstein condensation.
After introducing spectral zeta functions, we will illustrate one method often utilized to study the analytic continuation of such functions through an elementary example. We will then discuss recent results proving such an approach can be used in much more general settings, allowing the treatment of Dirac and non-self-adjoint operators appearing in many applications including gravitational backgrounds involving black holes or branes. After this, the relation to recent and ongoing regularization and high-energy asymptotics investigations will be discussed.