Quantum mechanics of complex Hamiltonian systems in one dimension

Abstract

With a view to obtaining further insight into the nature of eigenvalues and eigenfunctions of a stationary state one-dimensional Schrödinger equation corresponding to a non-Hermitian Hamiltonian H(x, p) we investigate the ground-state solutions for a variety of potentials within the framework of an extended complex phase space characterized by x = x1 + ip2, p = p1 + ix2, where (x1, p1) and (x2, p2) are real and considered as canonical pairs. The analyticity property of the eigenfunction alone is found sufficient to throw light on the nature of eigenvalues and eigenfunctions for different systems. It is noted that the imaginary part of the eigenvalue, Ei, turns out to be zero for all potentials V(x) with real couplings whereas it turns out to be nonzero for the case when the couplings are complex. The prescription is also extended to study the excited states. The problems related to the normalization of the eigenfunction and the boundary conditions to be used within this framework are also discussed.