PT-Symmetric Quantum Mechanics: A Precise and Consistent Formulation
  The physical condition that the expectation values of physical observables are real quantities is used to give a precise formulation of PT-symmetric quantum mechanics. A mathematically rigorous proof is given to establish the physical equivalence of PT-symmetric and conventional quantum mechanics. The results reported in this paper apply to arbitrary PT-symmetric Hamiltonians with a real and discrete spectrum. They hold regardless of whether the boundary conditions defining the spectrum of the Hamiltonian are given on the real line or a complex contour. 
Ali Mostafazadeh