Periodic orbit quantization

Main J, Mandelshtam V A, Wunner G and Taylor H S 1998 Harmonic inversion as a general method for periodic orbit quantization Nonlinearity1015... 

Periodic Orbit Quantization by Harmonic Inversion of Gutzwiller s Recurrence Function. 

Periodic-Orbit Quantization in Fully Chaotic Regime... 

The derivation of (4.13) shows that the equilibrium point quantization and the periodic-orbit quantization can be compared term by term. This comparison shows that the periodic-orbit quantization is able to take into account the anharmonicities in the direction of symmetric stretch. However, the anhar-monicities are neglected in the other directions transverse to the periodic orbit. Their full treatment requires the calculation of h corrections to the Gutzwiller trace formula, as shown elsewhere [14]. 

It should be noted that this periodic-orbit quantization is no longer valid in... 

The periodic-orbit quantization can be used to calculate not only the resonances but also the full shape of the photoabsorption cross section using (2.26) and (2.27). This semiclassical formula for the cross section separates in a natural way the smooth background from the oscillating structures due to the periodic orbits. In this way, the observation of emerging periodic orbits by the Fourier transform of the vibrational structures on top of the continuous absorption bands can be explained. 

Differences between the lifetimes obtained from equilibrium point quantization and periodic-orbit quantization appear as the bifurcation is approached. The lifetimes are underestimated by equilibrium point quantization but overestimated by periodic-orbit quantization. The reason for the upward deviation in the case of periodic-orbit quantization is that the Lyapunov exponent vanishes as the bifurcation is approached. The quantum eigenfunctions, however, are not characterized by the local linearized dynamics but extend over larger distances that are of more unstable character. 

Figure 13. Scattering resonances of a two-degree-of-freedom collinear model of the dissociation of Hgl2 [10], The filled dots are obtained by wavepacket propagation, the crosses by equilibrium point quantization with (2.8), the dotted circles from periodic-orbit quantization with (4.16). The solid lines are the curves corresponding to the Lyapunov exponents, Im E = -(A/2)Xp(Re ), of the fundamental periodic orbits p = 0, 1,2. The dashed line is the spectral gap, Im = ftP(l/2 Ref). The long-short dashed line is the curve corresponding to the escape rate, Im E = (h/2)P( Re ).

The periodic-orbit quantization based on the zeta function quantization... 

Recently, Hiipper et al. [160] have carried out a periodic-orbit quantization of the model of Ref. [159], as well as of another model [161] which predicts a shorter lifetime of about 3 fs instead of 6 fs. 

P. Gaspard To answer the question by Prof. Marcus, let me say that we have observed, in particular in Hgl2, that higher order perturbation theory around the saddle equilibrium point of the transition state may indeed be used to predict with a good accuracy the resonances just above the saddle. However, deviations appear for higher resonances and periodic-orbit quantization then turns out to be in better agreement than equilibrium point quantization. 

P. Gaspard To answer the question by Prof. Poliak, we expect from our present knowledge that the periodic-orbit quantization of the H + H2 dissociative dynamics on the Karplus-Porter surface can be performed with the same theory as applied to Hgl2. 

Cvitanovic, P. and Eckhardt, B. (1989). Periodic orbit quantization of chaotic systems, Phys. Rev. Lett. 63, 823-826. 

---