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.     

Development of an approximate method for quantum optical models and their pseudo-Hermicity

An approximate method is suggested to obtain analytical expressions for the eigenvalues and eigenfunctions of the some quantum optical models. The method is based on the Lie-type transformation of the Hamiltonians. In a particular case it is demonstrated that E × ɛ Jahn-Teller Hamiltonian can easily be solved within the framework of the suggested approximation. The method presented here is conceptually simple and can easily be extended to the other quantum optical models. We also show that for a purely imaginary coupling the E × ɛ Hamiltonian becomes non-Hermitian but Pσ₀-symmetric. Possible generalization of this approach is outlined. Presented at the 3rd International Workshop “Pseudo-Hermitian Hamiltonians in Quantum Physics”, Istanbul, Turkey, June 20–22, 2005.