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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 instatistical mechanics. Despite being non-Hermitian, this Hamiltonian appears topossess a real, positive spectrum. In the LDE, as in perturbation theory, theeigenvalues are naturally real, so a proof of this property devolves on theconvergence of the expansion. A proof of convergence of a modified version ofthe LDE is provided for the $ix^3$ potential in zero dimensions. The methodsdeveloped in zero dimensions are then extended to quantum mechanics, where weprovide numerical evidence for convergence.Comment: 26 pages (revtex), 8 figures (eps). Submitted to Phys. Rev.

physical review. d. particles, fields, gravitation, and cosmology/physical review. d. particles and fields

Applying the linear δ expansion to the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>i</mml:mi><mml:mrow><mml:msup><mml:mrow><mml:mi>φ</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math>interaction

Applying the linear δ expansion to theiφ3interaction