1. Introduction

Functional determinants of second-order differential operators are of great importance in many different fields. In physics, functional determinants provide the one-loop approximation to quantum field theories in the path integral formulation [21; 48]. In mathematics they describe the analytical torsion of a manifold [47].

Although there are various ways to evaluate functional determinants, the zeta function scheme seems to be the most elegant technique to use [9; 16; 17; 31]. This is the method introduced by Ray and Singer to define analytical torsion [47]. In physics its origin goes back to ambiguities in dimensional regularization when applied to quantum field theory in curved spacetime [11; 29].

For many second-order ordinary differential operators surprisingly simple answers can be given. The determinants for these situations have been related to boundary values of solutions of the operators, see, e.g., [8; 10; 12; 22; 23; 26; 36; 39; 40]. Recently, these results have been rederived with a simple and accessible method which uses contour integration techniques [33; 34; 35]. The main advantage of this approach is that it can be easily applied to general kinds of boundary conditions [35] and also to cases where the operator has zero modes [34; 35]; see also [37; 38; 42]. Equally important, for some higher dimensional situations the task of finding functional determinants remains feasible. Once again closed answers can be found but compared to one dimension technicalities are significantly more involved [13; 14]. It is the aim of this article to choose specific higher dimensional examples where technical problems remain somewhat confined. The intention is to illustrate that also for higher dimensional situations closed answers can be obtained which are easily evaluated numerically.