Maslov index

Coherent states and diverse semiclassical approximations to molecular wavepackets are essentially dependent on the relative phases between the wave components. Due to the need to keep this chapter to a reasonable size, we can mention here only a sample of original works (e.g., [202-205]) and some summaries [206-208]. In these, the reader will come across the Maslov index [209], which we pause to mention here, since it links up in a natural way to the modulus-phase relations described in Section III and with the phase-fiacing method in Section IV. The Maslov index relates to the phase acquired when the semiclassical wave function haverses a zero (or a singularity, if there be one) and it (and, particularly, its sign) is the consequence of the analytic behavior of the wave function in the complex time plane. 

In the three-branch horseshoe, the periodic oibit 0 is hyperbolic with reflection and has a Maslov index equal to no = 3 while the off-diagonal orbits 1 and 2 are hyperbolic without reflection with the Maslov index n = 2 [10]. Fitting of numerical actions, stability eigenvalues, and rotation numbers to polynomial functions in E can then be used to reproduce the analytical dependence on E. The resonance spectrum is obtained in terms of the zeros of (4.16) in the complex energy surface. 

Cartesian coordinates and momenta, the kt are quantum numbers, and m( is an even integer known as the Maslov index, [51] 2, for vibrational motion. Most successful applications of Eq. (3.36) require quasi-periodic motion... 

In the last equation, the integral transformation to initial variable representation is carried out. H denotes the total electronic and nuclear Hamiltonian and again i/ is the Maslov index at time t. Similarly, the transition amplitude from the initial total quantum state kiXi to the final nuclear position R at state k2, s described as... 

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