-symmetric potentials having continuous spectra

Abstract

One-dimensional -symmetric quantum-mechanical Hamiltonians having continuous spectra are studied. The Hamiltonians considered have the form H = p2 + V(x), where −∞ < x < ∞. The potential V(x) is odd in x, pure imaginary, and vanishes as |x| → ∞. Five -symmetric potentials are studied: the Scarf-II potential V1(x) = iA1 sech(x)tanh(x), which decays exponentially for large |x|; the rational potentials V2(x) = iA2x/(1 + x4) and V3(x) = iA3x/(1 + |x|3), which decay algebraically for large |x|; the step-function potential V4(x) = iA4 sgn(x)θ(2.5 − |x|), which has compact support; the regulated Coulomb potential V5(x) = iA5x/(1 + x2), which decays slowly as |x| → ∞ and thus may be viewed as a long-range potential. The real parameters An measure the strengths of these potentials. Numerical techniques for solving the time-independent Schrödinger eigenvalue problems associated with these potentials reveal that, in general, the spectra of these Hamiltonians 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 +∞. The complex eigenvalues occur in discrete complex-conjugate pairs and for Vn(x) (1 ⩽ n ⩽ 4) the number of these pairs is finite and increases as the value of the strength parameter An increases. However, for V5(x) there is an infinite sequence of discrete eigenvalues with a limit point at the origin. While this sequence is complex, it resembles the Balmer series for the hydrogen atom because it has inverse-square convergence.