Feshbach-type resonances

In the previous sections, we introduced resonance states and discussed situations in which resonances can be observed. In this section, we address the question of the origin for the appearance of resonances, or in other words, the basic question is what can bring about the formation of metastable states. In a very general manner, it is common to classify resonances into two main groups shape-type resonances and Feshbach-type resonances. Although the classification is not unique and may depend on the chosen representation of the Hamiltonian [46, 47], it can be extremely helpful in understanding the physical mechanism that leads to the formation of the metastable state. 

Feshbach-type resonances [51], also known as Fano resonances [52] and Floquet resonances [22] depending on the system studied, are formed in a different manner. We encounter this type of metastable states whenever a bound system is coupled to an external continuum. In the same spirit as before, one can define a reference Hamiltonian in which the closed channel containing the bound states is uncoupled from the open channel through which the asymptote can be reached. When the coupling is introduced, the previously bound state decays into the continuum of the open channel. The distinction from shape-type resonances, described above, is that the resonance state decays into a different channel of the reference Hamiltonian. 

The case of decay by energy transfer akin to Feshbach-type resonance(s) in collision theory, presents a probe of greater sensitivity to bound-state dynamics. This is clearly exemplified by the doorway channel model.69 In this model, the bound manifold is coupled to a dissociative manifold via a single ( doorway ) channel. This forces the system, if excited to some arbitrary state, to diffuse to the subspace spanned by the doorway channel in order to dissociate. 

Ho also computed series of high-lying Feshbach-type resonances converging to the TV = 3,..,9 hydrogen threshols (133). Several tens of resonances were reported. Chen (134) applied a basis set of B-spline functions within the saddle-point complex rotation method to compute parameters of twelve and low lying Feshbach resonances. 

The calculation of a Feshbach shape resonance has been carried out for a 2D superlattice of carbon nanotubes of period Ap on a 2D x,y plane shown in Fig. 4. The electronic structure is similar to the case of a superlattice of stripes [93-96,102] and this type of heterostructures at atomic limit can be classified as superlattices of quantum wires". While the charge carriers move as free charges in the x direction, the wire direction, they have to overcome a periodic potential barrier V(x,y), with period Ap, amplitude Vb... 

The right panel of Figure 2.9 shows that the rate of level v = 8 reaches zero twice. This multiple occurrence of the zero-width phenomenon has been related to the possibility to produce several times a diabatic-adiabatic coincidence as the intensity increases [69]. This is so because the adiabatic (vibrational) levels goes up faster than the diabatic ones and therefore a given adiabatic level can cross several diabatic levels as the intensity increases. It is to be noted that resonance coalescence, i.e., the existence of an (EP), requires an appropriate choice of both frequency and intensity, while a ZWR would show up at some critical intensity(ies), irrespective of the choice of the wavelength, for all resonances of Feshbach type. One must keep in mind that the classification into shape and Feshbach depends strongly on the wavelength. 

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... 

Resonances of the type illustrated in Figure 12.2 are called Feshbach resonances (Child 1974 ch.4 Fano and Rao 1986 ch.8 see also Figure 12.5). The quasi-bound states trapped by the Vn(.R) potential can only decay via coupling to the lower vibrational state because asymptotically the n = 1 channel is closed and therefore cannot be populated. This is different from the dissociation of CH30N0(Si), for example, [see Figure 7.10(a)] where the resonances can either decay via tunneling or alternatively by nonadiabatic coupling to the lower states. 

At higher energies, other types of resonances are found in these molecules. In ethylene, narrow Feshbach resonances above... 

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