Real Eigenvalue of a Non-Hermitian Hamiltonian System

Abstract

With a view to getting further insight into the solutions of one-dimensional analogous Schr?dinger equation for a non-hermitian (complex) Hamiltonian system, we investigate the quasi-exact PT - symmetric solutions for an octic potential and its variant using extended complex phase space approach characterized by x=x 1 +ip 2 , p=p 1 +ix 2 , where ( x 1 , p 1 ) and ( x 2 , p 2 ) are real and considered as canonical pairs. Besides the complexity of the phase space, complexity of potential parameters is also considered. The analyticity property of the eigenfunction alone is found sufficient to throw light on the nature of eigenvalue and eigenfunction of a system. The imaginary part of energy eigenvalue of a non-hermitian Hamiltonian exist for complex potential parameters and reduces to zero for real parameters. However, in the present work, it is found that imaginary component of the energy eigenvalue vanishes even when potential parameters are complex, provided that PT -symmetric condition is satisfied. Thus PT - symmetric version of a non-hermitian Hamiltonian possesses the real eigenvalue.