General properties of the spectrum of complex scaled Hamiltonians: Detachment point and localization threshold

In this paper we concentrate on the most prominent points describing string curves: the detachment point and the localization threshold. We show that the detachment point is identical to the potential height for symmetric potentials and to the lowest barrier height in the general asymmetric case. Furthermore, we show that the existence of a detachment point is due to spectral concentration at the barrier top. Concerning the localization threshold, we show that the localization properties alter drastically at this point; the classical mechanism of a particle being trapped between potential barriers breaks down at this point. We then derive a condition for the existence of the localization threshold in the framework of the Wenzel-Kramers-Brillouin method and consider the dependence of the localization threshold on physical potential parameters; it is shown that there is no simple correspondence of the localization threshold to such parameters, since the localization threshold also depends on the explicit analytical form of the potential