Pseudo-Hermitian Hamiltonians: tale of two potentials

For the non-Hermitian, PT-symmetric potentials V = m ² x ² + gx ²(ix)^ν with ν = 1 and 2, we construct the Q operator which gives both the positive-definite metric and an equivalent Hermitian Hamiltonian h. For the case ν = 1, where the theory may be defined on the real axis, h is reasonable but complicated. For the case ν = 2, where the theory must initially be defined on a contour in the complex x plane, we first introduce a real parametrization of the contour, and then calculate Q and h as an expansion in an angle θ. Theresultant h has less desirable properties. However, Q is not uniquely determined, and it may be possible to exploit this ambiguity to produce a more acceptable equivalent Hamiltonian. Presented at the 3rd International Workshop “Pseudo-Hermitian Hamiltonians in Quantum Physics”, Istanbul, Turkey, June 20–22, 2005.