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Equivalence of a complex<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi mathvariant="script">P</mml:mi><mml:mi mathvariant="script">T</mml:mi></mml:math>-symmetric quartic Hamiltonian and a Hermitian quartic Hamiltonian with an anomaly
In a recent paper Jones and Mateo used operator techniques to show that the non-Hermitian $\mathcal{P}\mathcal{T}$-symmetric wrong-sign quartic Hamiltonian $H=\frac{1}{2}{p}^{2}\ensuremath{-}g{x}^{4}$ has the same spectrum as the conventional Hermitian Hamiltonian $\stackrel{\texttildelow{}}{H}=\frac{1}{2}{p}^{2}+4g{x}^{4}\ensuremath{-}\sqrt{2g}x$. Here, this equivalence is demonstrated very simply by means of differential-equation techniques and, more importantly, by means of functional-integration techniques. …