  The linear $\delta$ expansion (LDE) is applied to the Hamiltonian $H=(p^2
+m^2 x^2)/2 + igx^3$, which arises in the study of Lee-Yang zeros in
statistical mechanics. Despite being non-Hermitian, this Hamiltonian appears to
possess a real, positive spectrum. In the LDE, as in perturbation theory, the
eigenvalues are naturally real, so a proof of this property devolves on the
convergence of the expansion. A proof of convergence of a modified version of
the LDE is provided for the $ix^3$ potential in zero dimensions. The methods
developed in zero dimensions are then extended to quantum mechanics, where we
provide numerical evidence for convergence.
