  Heat-kernel expansion and zeta function regularisation are discussed for
Laplace type operators with discrete spectrum in non compact domains. Since a
general theory is lacking, the heat-kernel expansion is investigated by means
of several examples. It is pointed out that for a class of exponential
(analytic) interactions, generically the non-compactness of the domain gives
rise to logarithmic terms in the heat-kernel expansion. Then, a meromorphic
continuation of the associated zeta function is investigated. A simple model is
considered, for which the analytic continuation of the zeta function is not
regular at the origin, displaying a pole of higher order. For a physically
meaningful evaluation of the related functional determinant, a generalised zeta
function regularisation procedure is proposed.
