The potential V (x) = −x 4 , which is unbounded below on the real line, can give rise to a well-posed bound state problem when x is taken on a contour in the lower-half complex plane. It is then P Tsymmetric rather than Hermitian. Nonetheless it has been shown numerically to have a real spectrum, and a proof of reality, involving the correspondence between ordinary differential equations and integrable systems, was subsequently constructed for the general class of potentials −(ix) N . For such Hamiltonians the natural P T metric is not positive definite, but a dynamically-defined positive-definite metric can be defined, depending on an operator Q. Further, with the help of this operator an equivalent Hermitian Hamiltonian h can be constructed. This programme has been carried out exactly for a few soluble models, and the first few terms of a perturbative expansion have been found for the potential m 2 x 2 + igx 3 . However, until now, the −x 4 potential has proved intractable. In the present paper we give explicit, closed-form expressions for Q and h, which are made possible by a particular parametrization of the contour in the complex plane on which the problem is defined. This constitutes an explicit proof of the reality of the spectrum. The resulting equivalent Hamiltonian has a potential with a positive quartic term together with a linear term.

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It can be shown using operator techniques that the non-Hermitian P Tsymmetric quantum mechanical Hamiltonian with a "wrong-sign" quartic potential −gx 4 is equivalent to a Hermitian Hamiltonian with a positive quartic potential together with a linear term. A naïve derivation of the same result in the path-integral approach misses this linear term. In a recent paper by Bender et al. it was pointed out that this term was in the nature of a parity anomaly and a more careful, discretized treatment of the path integral appeared to reproduce it successfully. However, on re-examination of this derivation we find that a yet more careful treatment is necessary, keeping terms that were ignored in that paper. An alternative, much simpler derivation is given using the additional potential that has been shown to appear whenever a change of variables to curvilinear coordinates is made in a functional integral.

Hierarchies of one-dimensional Hamiltonians in quantum mechanics admitting third-order differential ladder operators are studied. Each Hamiltonian has associated three-step Darboux (pseudo)-cycles and Painlevé IV equations as a closure condition. The whole hierarchy is generated applying some operations on the cycles. These operations are investigated in the frame of supersymmetric quantum mechanics and mainly involve algebraic manipulations. A consistent geometric representation for the hierarchy and cycles is built that also helps in understanding the operations. Three kinds of hierarchies are distinguished and a realization based on the harmonic oscillator Hamiltonian is supplied, giving an interpretation for the spectral properties of the Hamiltonians of each hierarchy.

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