  The field-theoretic one-loop effective action in a static scalar background
depending nontrivially on a single spatial coordinate is related, in the
proper-time formalism, to the trace of the evolution kernel (or heat kernel)
for an appropriate, one dimensional, quantum-mechanical Hamiltonian. We
describe a recursive procedure applicable to these traces for shape-invariant
Hamiltonians, resolving subtleties from the continuum mode contributions by
utilizing the expression for the regularized Witten index. For some cases which
include those of domain-wall-type scalar backgrounds, our recursive procedure
yields the full expression for the scalar or fermion one-loop effective action
in both (1+1) and (3+1)-dimensions.
