Transition states and the critical parameters of central potentials

Abstract

Transition states or quantum states of zero energy appear at the boundary between the discrete part of the spectrum of negative energies and the continuum part of positive energy states. As such, transition states can be regarded as a limiting case of a bound state with vanishing binding energy, emerging for a particular set of critical potential parameters. In this work we study the properties of these critical parameters for short range central potentials. To this end we develop two exact methods and also utilize the first and second order WKB approximations. Using these

Constraining the optical potential in the search for η-mesic 4He

“…Having excluded the optical potential predictions of unstable bound states in literature, we turn to examine the special case of an unstable state centered at zero energy. The case of a zero energy bound state (or zero energy resonance), sometimes referred to as a transition state [56] has been widely studied in literature [57] in the context of different physical situations and has also been observed in ultracold atoms [58]. Let us recall some basic facts: a bound state corresponds to a pole in the S-matrix for E < 0.…”

“…Ref. [56] has examined the occurrence of such states for a class of potentials of the form V (r, r 0 ) = − g r s f r r0 (g > 0, r 0 > 0), which include the Gaussian, exponential and Hulthen among others. For the Gaussian optical potential of the present work, we identify g with V 0 , s = 0 and f = exp(−r 2 /r 2 0 ).…”

“…Analytical as well as numerical solutions of the Schroedinger equation for these potentials are provided in Ref. [56]. It is shown that the existence of the transition state solution depends on a critical parameter given by and CS ηN amplitudes respectively (see Fig.…”

Local approximation to the critical parameters of quantum wells

“…Most of those bounds are given in terms of the potential-energy function. In a recent paper Liverts and Barnea [14] proceeded in a different way and proposed the calculation of the critical parameters for negative central-field quantum wells. To this end they applied two exact methods and the WKB approach, the latter for the estimation of the large-quantum number behaviour of the critical parameters.…”

“…In this context a critical parameter is the value of a potential parameter for which an energy eigenvalue is exactly zero (what the authors call a transition state). As they pointed out, one can obtain the exact number of bound states from the tables of critical parameters, as well as other relevant information about the eigenvalue equation [14].…”