Eigenvalue lattice

A. Quantum Eigenvalue Lattice and Good Spectroscopic Quantum Numbers... 

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.

Figure 20. Quantum eigenvalue lattices forji =22 = 5 and (a) hja = 0.2, (b) bja = 2. Solid and dashed lines are type / and type U relative equilibria, respectively. The large point at 7 = 0 is the overlapping projection of points C and D in Table 1. Those at = N are projections of points A and B.

Chapter 5 provides some examples of purely analyti( al tools useful for describing CA. It discusses methods of inferring cycle-state structure from global eigenvalue spectra, the enumeration of limit cycles, the use of shift transformations, local structure theory, and Lyapunov functions. Some preliminary research on linking CA behavior with the topological characteristics of the underlying lattice is also described. 

By the argument in Section IIB, the presence of a locally quadratic cylindrically symmetric barrier leads one to expect a characteristic distortion to the quantum lattice, similar to that in Fig. 1, which is confirmed in Fig. 7. The heavy lower lines show the relative equilibria and the point (0,1) is the critical point. The small points indicate the eigenvalues. The lower part of the diagram differs from that in Fig. 1, because the small amplitude oscillations of a spherical pendulum approximate those of a degenerate harmonic oscillator, rather than the fl-axis rotations of a bent molecule. Hence the good quantum number is... 

Figure 11. Quantum monodromy in the spectrum of the quadratic Hamiltonian of Eq. (38). The solid lines indicate relative equilibria. Filled circles mark the eigenvalues of the most stable isomer and those above the relevant effective potential barrier in Fig. 8. Open circles indicate interpenetrating eigenvalues of the secondary isomer. The transported unit cell moves over the hlled circle lattice, around the curved fold line connecting the two spectra.

In other words, in passing from the upper to the lower boundary in Fig. 14, as y increases, the point D pushes the comer of the low y lattice ahead of itself and forms the new boundary of the high y lattice in its wake. The eigenvalue reorganization between the limits y = 0 and y = 1 is therefore intrinsically tied up with quantum monodromy. 

Fig. 2.10. Eigenvalues of the Fourier component of the dipole-dipole interaction tensor in two-dimensional infinite lattices. The solid lines are for a triangular lattice, the dashed lines are for an analytical approximation (2.2.9), and the dotted lines are for a square lattice.

In eq. (18), Mi t) and Mzit) are the magnetizations at time t for sites 1 and 2 respectively, M oc) is the equilibrium value of the magnetization in either site, R is the spin-lattice relaxation rate (= 1/Ti) for the two sites, which are assumed to be equal, and k is the exchange rate. For a larger system, numerical solutions are readily available with standard eigenvalue methods [42-44]. 

Li is an atomic configuration of the site i, with probability p Li) in the GWF and po Li) in the HWF respectively, whereas L is a configuration of the remaining sites of the lattice. Note that this prescription does not change the phase of the wave function as the eigenvalues of the operators Ti are real. The correlations are local, and the configuration probabilities for different sites are independent. 

The dynamic RIS model developed for investigating local chain dynamics is further improved and applied to POE. A set of eigenvalues characterizes the dynamic behaviour of a given segment of N motional bonds, with v isomeric states available to each bond. The rates of transitions between isomeric states are assumed to be inversely proportional to solvent viscosity. Predictions are in satisfactory agreement with the isotropic correlation times and spin-lattice relaxation times from 13C and 1H NMR experiments for POE. 

Fig. 6. The ratios of the moduli of the two largest eigenvalues of the transition matrix, >,2 Ai, versus x, for chains on four-choice cubic lattice and three-choice square lattice. Curves 1, 2, and 3 represent chains with increasing sizes of the largest excluded polygons.

Fig. 7. Distribution of several consecutive eigenvalues starting with Xlt in complex plane for chains on four-choice cubic lattice. Outer circle (o) x= log 6. Inner circle ( ) x = log 0.4. Radii of the circles = X2. ...

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