PT-Symmetric Quantum Mechanics
  This paper proposes to broaden the canonical formulation of quantum mechanics. Ordinarily, one imposes the condition $H^\dagger=H$ on the Hamiltonian, where $\dagger$ represents the mathematical operation of complex conjugation and matrix transposition. This conventional Hermiticity condition is sufficient to ensure that the Hamiltonian $H$ has a real spectrum. However, replacing this mathematical condition by the weaker and more physical requirement $H^\ddag=H$, where $\ddag$ represents combined parity reflection and time reversal ${\cal PT}$, one obtains new classes of complex Hamiltonians whose spectra are still real and positive. This generalization of Hermiticity is investigated using a complex deformation $H=p^2+x^2(ix)^\epsilon$ of the harmonic oscillator Hamiltonian, where $\epsilon$ is a real parameter. The system exhibits two phases: When $\epsilon\geq0$, the energy spectrum of $H$ is real and positive as a consequence of ${\cal PT}$ symmetry. However, when $-1<\epsilon<0$, the spectrum contains an infinite number of complex eigenvalues and a finite number of real, positive eigenvalues because ${\cal PT}$ symmetry is spontaneously broken. The phase transition that occurs at $\epsilon=0$ manifests itself in both the quantum-mechanical system and the underlying classical system. Similar qualitative features are exhibited by complex deformations of other standard real Hamiltonians $H=p^2+x^{2N}(ix)^\epsilon$ with $N$ integer and $\epsilon>-N$; each of these complex Hamiltonians exhibits a phase transition at $\epsilon=0$. These ${\cal PT}$-symmetric theories may be viewed as analytic continuations of conventional theories from real to complex phase space. 
Carl Bender (Washington U.), Stefan Boettcher (Los Alamos and Clark   Atlanta U.), and Peter Meisinger (Washington U.)