Variational ansatz for PJ-symmetric quantum mechanics

A variational calculation of the energy levels of the class of PJ-invariant quantum mechanical models described by the non-Hermitian Hamiltonian H = p(2) - (ix)(N) with N positive and x complex is presented. The energy levels are determined by finding the stationary points of the functional [H](a,b,c) = (integral(C)dx psi(x) H psi(x))/(integral(C)dx psi(2)(x)), where psi(x) = (ix)(c)exp(a(ix)(b)) is a three-parameter class of PJ-invariant trial wave functions. The integration contour C used to define [H](a,b,c) lies inside a wedge in the complex-x plane in which the wave function falls off exponentially at infinity. Rather than having a local minimum the functional has a saddle point in the three-parameter (a,b,c)-space. At this saddle point the numerical prediction for the ground-state energy is extremely accurate for a wide range of N. The methods of supersymmetric quantum mechanics are used to determine approximate wave functions and energy eigenvalues of the excited states of this class of non-Hermitian Hamiltonians. (C) 1999 Published by Elsevier Science B.V. All rights reserved.