Semiclassical calculation of the C operator in PT-symmetric quantum mechanics

To determine the Hilbert space and inner product for a quantum theory defined by a non-Hermitian PT-symmetric Hamiltonian H, it is necessary to construct a new time-independent observable operator called C. It has recently been shown that for the cubic DT-symmetric Hamiltonian H = p(2) + x(2) + iis an element ofx(3) one can obtain C as a perturbation expansion in powers of is an element of. This Letter considers the more difficult case of noncubic Hamiltonians of the form H = p(2) + x(2) (ix)(delta) (delta greater than or equal to 0). For these Hamiltonians it is shown how to calculate C by using nonperturbative semiclassical methods. (C) 2004 Elsevier B.V. All rights reserved.