  We present a method for evaluating the partition function in a varying
external field. Specifically, we look at the case of a non-interacting,
charged, massive scalar field at finite temperature with an associated chemical
potential in the background of a delta-function potential. Whilst we present a
general method, valid at all temperatures, we only give the result for the
leading order term in the high temperature limit. Although the derivative
expansion breaks down for inhomogeneous backgrounds we are able to obtain the
high temperature expansion, as well as an analytic expression for the zero
point energy, by way of a different approximation scheme, which we call the
{\it local Born approximation} (LBA).

High Energy Physics - Theory;High Energy Physics - Phenomenology