Gutzwiller trace formula

Rather than looking at the spectrum obtained from the secular determinant (5), we will here consider the spectrum SG for fixed wavenumber k and than average over k. One can write the spectrum in terms of a periodic orbit trace formula reminiscent to the celebrate Gutzwiller trace formula being a semiclassical approximation of the trace of the Green function (Gutzwiller 1990). We write the density of states in terms of the traces of SG, that is,... 

C. Gutzwiller Trace Formula for Isolated Periodic Orbits... 

The highly excited and reactive dynamics, the details of which have been made accessible by recently developed experimental techniques, are characterized by transitions between classically regular and chaotic regimes. Now molecular spectroscopy has traditionally relied on perturbation expansions to characterize molecular energy spectra, but such expansions may not be valid if the corresponding classical dynamics turns out to be chaotic. This leads us to a reconsideration of such perturbation techniques and provides the starting point for our discussion. From there, we will proceed to discuss the Gutzwiller trace formula, which provides a semiclassical description of classically chaotic systems. 

The periodic-orbit contribution derived by Gutzwiller is general and applies to different kinds of periodic orbits. However, the applicability of (2.13) rests on the property that the periodic orbits are isolated, that is, they do not belong to a continuous family. This is the case in hyperbolic dynamical systems where all the periodic orbits are linearly unstable. We should emphasize that the Gutzwiller trace formula may apply both to bounded and scattering systems. 

Gutzwiller trace formula should present a peak. At the leading order, the amplitude is predicted to have a divergence because some stability eigenvalues pass through = 1 so that the denominator vanishes in Eq. (2.13). Consequently, uniform semiclassical approximations are required in the vicinity of bifurcations, which show that the amplitude is strongly peaked but still remains finite. 

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... 

In fact, the latter is the leading contribution to Gutzwiller s trace formula (Gut-zwiller, 1990), namely the contribution of the two-bounce periodic orbit between the two spheres without repetition, with the action Spo(k) = 2(r—2d)k where 2 (r — 2a) is the length of the geometric path. Note that the semiclassical result is suppressed by a factor of 1/4 in comparison to the small-scatterer one. 

Moreover, new semiclassical methods have been developed that are based on the Gutzwiller and Berry-Tabor trace formulas [12, 13]. These methods allow the calculation of energy levels or quantum resonances in systems with many interfering periodic orbits, as is the case for chaotic dynamics. 

The purpose of this chapter is to review the recent results obtained in this context over the last decade and, in particular, in our group. The report is organized as follows. In Section II, we summarize the relevant quantum-mechanical principles and the Gutzwiller and Berry-Tabor trace formulas. 

As we discussed in Section II in relation to (2.41), a survival amplitude has a semiclassical behavior that is directly related to the periodic orbits by the Gutzwiller or the Berry-Tabor trace formulas, in contrast to the quasi-classical quantities (2.42) or (3.3). Therefore, we may expect the function (3.7) to present peaks on the intermediate time scale that are related to the classical periodic orbits. For such peaks to be located at the periodic orbits periods, we have to assume that die level density is well approximated as a sum over periodic orbits whose periods Tp = 3eSp and amplitudes vary slowly over the energy window [ - e, E + e]. A further assumption is that the energy window contains a sufficient number of energy levels. At short times, the semiclassical theory allows us to obtain... 

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. 

In order to exhibit the basic ideas of Gutzwiller s method, we follow an elegant derivation of the trace formula given by Miller (1975). We restrict ourselves to a two degree of freedom bounded autonomous system with Hamiltonian H. The spectrum of H is discrete and determined by... 

This is the trace formula, first derived in a physics context by Gutzwiller in 1971. Similar periodic orbit formulae were also derived in the context of abstract mathematical dynamical systems (Selberg (1956), McKean (1972)). The sum in (4.1.72) extends over all possible periodic orbits... 

The forward application of (4.1.72) is much more diflScult to perform, but has been accomplished in many cases. Gutzwiller (1971) was the first to apply the trace formula to a real quantum mechanical problem. It consists of electrons with an asymmetric mass tensor moving in a Coulomb potential. This problem is important in semiconductor physics. Gutzwiller was able to compute good approximations to the first few quantum eigenstates of this system. 

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... 

In (10.4.43) the scaled energy condition is assumed, i.e. the ratio of the total potential strength and the energy are constant. Therefore, using (10.4.43) in Gutzwiller s trace formula does not predict the usual level density p E), but the level density under the condition of a constant potential to energy ratio. We denote this modified level density by d E). It is given by... 

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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