Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry

Abstract

The condition of self-adjointness ensures that the eigenvalues of a Hamiltonian are real and bounded below. Replacing this condition by the weaker condition of ${\cal PT}$ symmetry, one obtains new infinite classes of complex Hamiltonians whose spectra are also real and positive. These ${\cal PT}$ symmetric theories may be viewed as analytic continuations of conventional theories from real to complex phase space. This paper describes the unusual classical and quantum properties of these theories.

Figures (3)

• Figure 1: Energy levels of the Hamiltonian $H=p^2-(ix)^N$ as a function of the parameter $N$. There are three regions: When $N\geq2$ the spectrum is real and positive. The lower bound of this region, $N=2$, corresponds to the harmonic oscillator, whose energy levels are $E_n=2n+1$. When $1<N<2$, there are a finite number of real positive eigenvalues and an infinite number of complex conjugate pairs of eigenvalues. As $N$ decreases from $2$ to $1$, the number of real eigenvalues decreases; when $N\leq1.42207$, the only real eigenvalue is the ground-state energy. As $N$ approaches $1^+$, the ground-state energy diverges. For $N\leq1$ there are no real eigenvalues.
• Figure 2: Wedges in the complex-$x$ plane containing the contour on which the eigenvalue problem for the differential equation (\ref{['e2']}) for $N=4.2$ is posed. In these wedges $\psi(x)$ vanishes exponentially as $|x|\to\infty$. The wedges are bounded by Stokes lines of the differential equation. The center of the wedge, where $\psi(x)$ vanishes most rapidly, is an anti-Stokes line.
• Figure 3: The $m\neq0$ analog of Fig. 1. Note that transitions occur at $N=2$ and $N=1$.