PT-symmetric potentials having continuous spectra

Abstract

One-dimensional PT-symmetric quantum-mechanical Hamiltonians having continuous spectra are studied. The Hamiltonians considered have the form $H=p^2+V(x)$, where $V(x)$ is odd in $x$, pure imaginary, and vanishes as $|x|\to\infty$. Five PT-symmetric potentials are studied: the Scarf-II potential $V_1(x)=iA_1\,{\rm sech}(x)\tanh(x)$, which decays exponentially for large $|x|$; the rational potentials $V_2(x)=iA_2\,x/(1+x^4)$ and $V_3(x)=iA_3\,x/(1+|x|^3)$, which decay algebraically for large $|x|$; the step-function potential $V_4(x)=iA_4\,{\rm sgn}(x)\theta(2.5-|x|)$, which has compact support; the regulated Coulomb potential $V_5(x)=iA_5\,x/(1+x^2)$, which decays slowly as $|x|\to\infty$ and may be viewed as a long-range potential. The real parameters $A_n$ measure the strengths of these potentials. Numerical techniques for solving the time-independent Schr\"odinger eigenvalue problems associated with these potentials reveal that the spectra of the corresponding Hamiltonians exhibit universal properties. In general, the eigenvalues are partly real and partly complex. The real eigenvalues form the continuous part of the spectrum and the complex eigenvalues form the discrete part of the spectrum. The real eigenvalues range continuously in value from $0$ to $+\infty$. The complex eigenvalues occur in discrete complex-conjugate pairs and for $V_n(x)$ ($1\leq n\leq4$) the number of these pairs is finite and increases as the value of the strength parameter $A_n$ increases. However, for $V_5(x)$ there is an {\it infinite} sequence of discrete eigenvalues with a limit point at the origin. This sequence is complex, but it is similar to the Balmer series for the hydrogen atom because it has inverse-square convergence.