Critical points quantum monodromy

It is, however, more revealing in the context of monodromy to allow/(s, ) to pass from one Riemann sheet to the next, at the branch cut, a procedure that leads to the construction in Fig. 4, due to Sadovskii and Zhilinskii [2], by which a unit cell of the quantum lattice, with sides defined here by unit changes in k and v, is transported from one cell to the next on a path around the critical point at the center of the lattice. Note, in particular, that the lattice is locally regular in any region of the [k, s) plane that excludes the critical point and that any vector in the unit cell such as the base vector, marked by arrows, rotates as the cell is transported around the cycle. Consequently, the transported dashed cell differs from that of the original quantized lattice. 

Figure 8. Evidence of quantum monodromy in the the spherical pendulum eigenvalue lattice. The heavy continuous lines are the relative equihbria, and the large dot indicates the critical point.

As mentioned earlier, the most interesting point, with regard to the quantum monodromy, is that the two critical points C and D in Table I now he in the singular section of fc2(7z. A).) at 713 = / = 0, which means that the form with a single conical point in the right-hand panel of Fig. 15 goes over to one with two such points in Fig. 19, which is the case denoted as i 11 r in Tables II and IE of Grondin et al. [3]. 

It is evident that the projections of the fixed points A and B, in the EJ map, always lie on relative equilibria of type I, but that the position of the overlapping projections of C and D depends on the sign of b — a. If > b the double point is isolated between the two type I equilibria, and quantum monodromy is expected, for a sufficiently dense quantum lattice. If, on the other hand, the critical point lies on the type II relative equilibrium line and... 

It was shown in the earlier sections that the existence or nonexistence of quantum monodromy in two-dimensional maps depends on the relative dispositions of the critical points and relative equilibria of the Hamiltonian, which involves a search for the stationary points of with respect to K and Qk- For L = 0 there is a root at the critical point K = J, and other possible roots given by... 

The second type of quantum monodromy occurs in the computed bending-vibrational bands of LiCN/LiNC, in which the role of the isolated critical point is replaced by that of a finite folded region of the spectrum, where the vibrational states of the secondary isomer LiNC interpenetrate those of LiCN [9, 10]. The folded region is finite in this case, because the secondary minimum on the potential surface merges with the transition state as the angular momentum increases. However, the shape of the potential energy surface in HCN prevents any such angular momentum cut-off, so monodromy is forbidden [10]. 

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