Feshbach resonances infinite

Probably the most accurate positron-hydrogen s-wave phase shifts are those obtained by Bhatia et al. (4974), who avoided the possibility of Schwartz singularities by using a bounded variational method based on the optical potential formalism described previously. These authors chose their basis functions spanning the closed-channel Q-space, see equation (3.44), to be of essentially the same Hylleraas form as those used in the Kohn trial function, equation (3.42), and their most accurate results were obtained with 84 such terms. By extrapolating to infinite u in a somewhat similar way to that described in equation (3.54), they obtained phase shifts which are believed to be accurate to within 0.0002 rad. They also established that there are no Feshbach resonances below the positronium formation threshold. 

Feshbach resonances is purely model dependent since trapping well may exist on one type of adiabatic potential, say in hyperspherical coordinates, while only a barrier may exist on another type, say in natural collision coordinates. However, this is not correct since there are fundamental differences between QBS and Feshbach states. First, the pole structure of the S-matrix is intrinsically different in the two cases. A Feshbach resonance corresponds to a single isolated pole of the scattering matrix (S-matrix) below the real axis of the complex energy plane, see the discussion below. On the other hand, the barrier resonance corresponds to an infinite sequence of poles extending into the lower half plane. For a parabolic barrier, it is easy to show that the pole positions are given by... 

Infinite series of Feshbach resonances due to the Coulomb tail... 

Photoionization of neutral atoms and molecules and electron-ion collisions, for example, are rich in infinite Rydberg series of Feshbach resonances. On the other hand, only a finite number of Feshbach (and possibly shape) resonances occur in electron-neutral collisions and photodetachment of an electron attached to a neutral species, with an exception of the following cases. 

Figure 4.13 Adiabatic hyperspherical potentials, Eq. (101), in a.u. without the adiabatic correction term for He of symmetries 1,3SC and 1,3P° converging to the asymptotic limit e + He+(n = 2), plotted against the hyperradius R in a.u. Each potential supports an infinite number of Rydberg states of Feshbach resonance, of which the lowest level is indicated by a horizontal line. Figure from Ref. [90], Note the difference in notation.

According to Eq. (81) with /S = 6, the three a values for H in Figure 4.14 are —3.708, 2.000, and 9.708. Thus, the asymptotically lowest hyperspherical potential supports an infinite series of Feshbach resonances in the nonrelativistic approximation, although only three lowest members remain as resonances after corrections for the relativistic and radiative effects [80, 82], as was mentioned in Section 3.1.2. Only the lowest member is indicated in the figure by a horizontal line. This resonance is supported by the diabatic potential with A = — 1 connecting from the lowest curve for large p to the middle curve for small p. 

Figure 4.15 Hyperspherical potentials without the adiabatic correction term for Hef P0) converging to the asymptotic limits e + He+(n = 4,5,6). Each potential supports an infinite Rydberg series of Feshbach resonances. Some of them exhibit typical cases of inter-series overlapping resonances illustrated in Figures 4.9 and 4.10. Figure from Ref. [69].

Here, all the three particles are assumed to have a unit charge. The value of j3 is 6 for H with an infinitely heavy nucleus and is 8 for Ps. With the values of a from these analytic formulas, and hence f, Eq. (80) played an essential role in the analysis of the Feshbach series of resonances in muonic... 

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