On two complementary approaches aiming at the definition of the determinant of an elliptic partial differential operator

TL;DR

This paper unifies two different approaches—mathematical and physical—for defining the determinant of elliptic differential operators with complex spectra using zeta functions, providing a clear bridge between them.

Contribution

It explicitly connects the mathematical and physical methods for defining determinants via zeta functions, clarifying their convergence and providing an explicit bridging formula.

Findings

Unified the mathematical and physical approaches to operator determinants

Derived an explicit formula linking the two methods

Clarified conditions for defining zeta functions and determinants

Abstract

We bring together two apparently disconnected lines of research (of mathematical and of physical nature, respectively) which aim at the definition, through the corresponding zeta function, of the determinant of a differential operator possessing, in general, a complex spectrum. It is shown explicitly how the two lines have in fact converged to a meeting point at which the precise mathematical conditions for the definition of the zeta function and the associated determinant are easy to understand from the considerations coming up from the physical approach, which proceeds by stepwise generalization starting from the most simple cases of physical interest. An explicit formula that establishes the bridge between the two approaches is obtained.