This paper explores the properties of non-Hermitian Hamiltonians with PT symmetry, which are complex Hamiltonians whose spectra are real and positive. The authors demonstrate that PT symmetry ensures the reality of the spectrum, even though the Hamiltonian is not Hermitian. They examine the Hamiltonian $ H = p^2 + m^2x^2 - (ix)^N $, where $ N $ is real, and show that as $ N $ varies, the spectrum transitions from real and positive to complex. The paper discusses the classical and quantum properties of these Hamiltonians, including the behavior of energy levels and the role of PT symmetry in maintaining the reality of the spectrum. The authors also analyze the Schrödinger equation associated with this Hamiltonian and show that the boundary conditions must be extended into the complex plane for $ N \neq 2 $. They use numerical and asymptotic methods to study the spectrum and find that for $ N \geq 2 $, the spectrum is real and positive, while for $ 1 < N < 2 $, the spectrum becomes complex. At $ N = 1 $, there are no real eigenvalues. The paper also discusses the implications of these results for quantum field theories and the behavior of eigenvalues near $ N = 1 $ and $ N = 2 $. The authors conclude that PT-symmetric Hamiltonians have unique properties, including the possibility of real eigenvalues even in non-Hermitian systems, and that these systems can be used to study various physical phenomena, such as delocalization transitions and population biology. The paper also highlights the importance of PT symmetry in maintaining the reality of the spectrum and the role of complex contours in the analysis of these systems.This paper explores the properties of non-Hermitian Hamiltonians with PT symmetry, which are complex Hamiltonians whose spectra are real and positive. The authors demonstrate that PT symmetry ensures the reality of the spectrum, even though the Hamiltonian is not Hermitian. They examine the Hamiltonian $ H = p^2 + m^2x^2 - (ix)^N $, where $ N $ is real, and show that as $ N $ varies, the spectrum transitions from real and positive to complex. The paper discusses the classical and quantum properties of these Hamiltonians, including the behavior of energy levels and the role of PT symmetry in maintaining the reality of the spectrum. The authors also analyze the Schrödinger equation associated with this Hamiltonian and show that the boundary conditions must be extended into the complex plane for $ N \neq 2 $. They use numerical and asymptotic methods to study the spectrum and find that for $ N \geq 2 $, the spectrum is real and positive, while for $ 1 < N < 2 $, the spectrum becomes complex. At $ N = 1 $, there are no real eigenvalues. The paper also discusses the implications of these results for quantum field theories and the behavior of eigenvalues near $ N = 1 $ and $ N = 2 $. The authors conclude that PT-symmetric Hamiltonians have unique properties, including the possibility of real eigenvalues even in non-Hermitian systems, and that these systems can be used to study various physical phenomena, such as delocalization transitions and population biology. The paper also highlights the importance of PT symmetry in maintaining the reality of the spectrum and the role of complex contours in the analysis of these systems.