Functional determinants for radial operators

TL;DR

This paper extends the Gel'fand-Yaglom method to compute functional determinants of radially separable operators in higher dimensions, providing simpler formulas that agree with existing methods and facilitate numerical calculations.

Contribution

It introduces new, simplified expressions for functional determinants of radial operators in multiple dimensions, generalizing previous one-dimensional results.

Findings

Derived formulas agree with angular momentum cutoff and Feynman diagram approaches.

Simplified expressions facilitate numerical evaluation of determinants.

Results applicable across various dimensions.

Abstract

We derive simple new expressions, in various dimensions, for the functional determinant of a radially separable partial differential operator, thereby generalizing the one-dimensional result of Gel'fand and Yaglom to higher dimensions. We use the zeta function formalism, and the results agree with what one would obtain using the angular momentum cutoff method based on radial WKB. The final expression is numerically equal to an alternative expression derived in a Feynman diagrammatic approach, but is considerably simpler.