The paper by Carl M. Bender and Stefan Boettcher explores the properties of non-Hermitian Hamiltonians that exhibit $\mathcal{PT}$ symmetry, a weaker condition than self-adjointness. $\mathcal{PT}$ symmetry ensures that the eigenvalues of these Hamiltonians are real and positive, even though the Hamiltonians themselves are not Hermitian. The authors focus on the Hamiltonian \( H = p^2 + m^2 x^2 - (i x)^N \) and investigate its spectrum as a function of the parameter \( N \) and mass \( m^2 \). Key findings include: 1. **Spectrum Behavior**: For \( N \geq 2 \), the spectrum is infinite, discrete, and entirely real and positive. At \( N = 2 \), a phase transition occurs, where the spectrum becomes complex for \( 1 < N < 2 \), and the ground-state energy diverges as \( N \to 1^+ \). 2. **$\mathcal{PT}$ Symmetry and Spectrum Positivity**: $\mathcal{PT}$ symmetry is crucial for the positivity of the spectrum. Hamiltonians that are not $\mathcal{PT}$-symmetric have complex spectra. 3. **Classical and Quantum Properties**: The classical motion of a particle described by these Hamiltonians exhibits interesting behavior, such as the merging of energy levels and the infinite period of motion at \( N = 2 \). 4. **Applications**: Non-Hermitian $\mathcal{PT}$-invariant Hamiltonians have applications in various fields, including condensed matter physics and quantum field theory, where they can induce delocalization transitions and affect the behavior of systems like vortex fluxline depinning in superconductors. The authors use numerical and asymptotic methods to study the Hamiltonian, providing insights into the fundamental properties of $\mathcal{PT}$-symmetric theories.The paper by Carl M. Bender and Stefan Boettcher explores the properties of non-Hermitian Hamiltonians that exhibit $\mathcal{PT}$ symmetry, a weaker condition than self-adjointness. $\mathcal{PT}$ symmetry ensures that the eigenvalues of these Hamiltonians are real and positive, even though the Hamiltonians themselves are not Hermitian. The authors focus on the Hamiltonian \( H = p^2 + m^2 x^2 - (i x)^N \) and investigate its spectrum as a function of the parameter \( N \) and mass \( m^2 \). Key findings include: 1. **Spectrum Behavior**: For \( N \geq 2 \), the spectrum is infinite, discrete, and entirely real and positive. At \( N = 2 \), a phase transition occurs, where the spectrum becomes complex for \( 1 < N < 2 \), and the ground-state energy diverges as \( N \to 1^+ \). 2. **$\mathcal{PT}$ Symmetry and Spectrum Positivity**: $\mathcal{PT}$ symmetry is crucial for the positivity of the spectrum. Hamiltonians that are not $\mathcal{PT}$-symmetric have complex spectra. 3. **Classical and Quantum Properties**: The classical motion of a particle described by these Hamiltonians exhibits interesting behavior, such as the merging of energy levels and the infinite period of motion at \( N = 2 \). 4. **Applications**: Non-Hermitian $\mathcal{PT}$-invariant Hamiltonians have applications in various fields, including condensed matter physics and quantum field theory, where they can induce delocalization transitions and affect the behavior of systems like vortex fluxline depinning in superconductors. The authors use numerical and asymptotic methods to study the Hamiltonian, providing insights into the fundamental properties of $\mathcal{PT}$-symmetric theories.