Pseudo-Hermitian description ofPT-symmetric systems defined on a complex contour

We describe a method that allows for a practical application of the theory of pseudo-Hermitian operators to PT -symmetric systems defined on a complex contour. We apply this method to study the Hamiltonians H = p 2 + x 2 (i x ) ν with ν (−2, ∞ ) that are defined along the corresponding anti-Stokes lines. In particular, we reveal the intrinsic non-Hermiticity of H for the cases that ν is an even integer, so that H = p 2 ± x 2+ν , and give a proof of the discreteness of the spectrum of H for all ν (−2, ∞ ). Furthermore, we study the consequences of defining a square-well Hamiltonian on a wedge-shaped complex contour. This yields a PT -symmetric system with a finite number of real eigenvalues. We present a comprehensive analysis of this system within the framework of pseudo-Hermitian quantum mechanics. We also outline a direct pseudo-Hermitian treatment of PT -symmetric systems defined on a complex contour which clarifies the underlying mathematical structure of the formulation of PT -symmetric quantum mechanics based on the charge-conjugation operator. Our results provide conclusive evidence that pseudo-Hermitian quantum mechanics provides a complete description of general PT -symmetric systems regardless of whether they are defined along the real line or a complex contour.