Quantization Gutzwiller trace formula

We will start by describing the relevant aspects of the classical dynamics, introducing the concepts of the invariant set and the repeller, the Smale horseshoe and its symbolic dynamics, and the bifurcations at their origin [19]. We then turn to the semiclassical quantization based on the Gutzwiller trace formula and the zeta functions. We proceed to show how this new theoretical framework allows us to explain the distribution of resonances in several molecules like Hgl2, CO2, and H3, to calculate their lifetimes, and to provide a synthesis with respect to previous work. 

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]. 

About 50 years after Einstein, Gutzwiller applied the path integral method with a semiclassical approximation and succeeded to derive an approximate quantization condition for the system that has fully chaotic classical counterpart. His formula expresses the density of states in terms of unstable periodic orbits. It is now called the Gutzwiller trace formula [9,10]. In the last two decades, several physicists tested the Gutzwiiler trace formula for various... 

Quantum calculations for a classically chaotic system are extremely hard to perform. If more than just the ground state and a few excited states are required, semiclassical methods may be employed. But it was not before the work of Gutzwiller about two decades ago that a semiclassical quantization scheme became available that is powerful enough to deal with chaos. Gutzwiller s central result is the trace formula which is derived in Section 4.1.3. 

Currently available numerical results indicate that the one-dimensional heUum atom is completely chaotic. The best-known semiclassical quantization procedure for completely chaotic systems is Gutzwiller s trace formula (see Section 4.1.3), which is based on classical periodic orbits. Therefore we search for simple periodic orbits of the one-dimensional he-hum atom. Since a two-electron orbit is periodic if the orbits ni t), 0i t)) and (ri2(t), 2( )) of the first and second electron have a common period, the periodic orbits of the one-dimensional model can be labelled with two integers, m and n, which count the 27r-multiplicity of the angle variables 0i and 02 after completion of the orbit. Therefore, if for some periodic orbit... 

Quantized chaos, or quantum chaology (see Section 4.1), is about understanding the quantum spectra and wave functions of classically chaotic systems. The semiclassical method is one of the sharpest tools of quantum chaology. As discussed in Section 4.1.3 the central problem of computing the semiclassical spectrum of a classically chaotic system was solved by Gutzwiller more than 20 years ago. His trace formula (4.1.72) is the basis for all semiclassical work on the quantization of chaotic systems. 

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