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,... 

Trace formulas like (8) are a starting point for analysing the statistical properties of quantum spectra. The statistical quantities such as the two-point correlation function can be written in terms of the density of states d(9,N), that is,... 

In fact, with the help of Krein s trace formula, the quantum field theory calculation is mapped onto a quantum mechanical billiard problem of a point-particle scattered off a finite number of non-overlapping spheres or disks i.e. classically hyperbolic (or even chaotic) scattering systems. 

For more complicated geometries, the computations become more and more involved as it is the case for the ordinary electromagnetic Casimir effect. However, Casimir calculations of a finite number of immersed nonoverlapping spherical voids or rods, i.e. spheres and cylinders in 3 dimensions or disks in 2 dimensions, are still doable. In fact, these calculations simplify because of Krein s trace formula (Krein, 2004 Beth and Uhlenbeck, 1937)... 

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. 

Vol. 1503 S. Shokranian, The Selberg-Arthur Trace Formula (Lectures by J. Arthur). VII, 97 pages. 1991. 

Since the birth of quantum theory, there has been considerable interest in the transition from quantum to classical mechanics. Because the two formulations are given in a different theoretical framework (nonlinear classical trajectories versus expectation values of linear operators), this transition is far more involved than the naive limit —> 0 suggests. By exploring the classical limit of quantum mechanics, new theoretical concepts have been developed, including path integrals [1], various phase-space representations of quantum mechanics [2], the semiclassical propagator and the trace formula [3], and the notion of quantum... 

Describing complex wave-packet motion on the two coupled potential energy surfaces, this quantity is also of interest since it can be monitored in femtosecond pump-probe experiments [163]. In fact, it has been shown in Ref. 126 employing again the quasi-classical approximation (104) that the time-and frequency-resolved stimulated emission spectrum is nicely reproduced by the PO calculation. Hence vibronic POs may provide a clear and physically appealing interpretation of femtosecond experiments reflecting coherent electron transfer. We note that POs have also been used in semiclassical trace formulas to calculate spectral response functions [3]. 

A. Time Evolution in Quantum Mechanics and Trace Formulas... 

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

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. 

The interrelations between the propagator, the resolvent, and the level density will be central to our discussion. In particular, the trace formulas referred to in Section I represent semiclassical approximations to the quantities (2.6) or (2.7) and turn out to involve the periodic orbits of the classical dynamics. 

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 amplitude of the periodic orbits is therefore determined by the linear stability with respect to perturbations transverse to the orbit. In this sense, the leading term in expression (2.13), obtained by setting C = 0, treats the dynamics transverse to the orbit at the level of the harmonic approximation. The nonlinear stability properties appear thus as anharmonic corrections to the dynamics transverse to the orbit. These anharmonicities contribute to the trace formula by corrections given in terms of series in powers of the Planck constant involving the coefficients C , which can be obtained as Feynman diagrams [14, 31]. 

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. 

In this section, we arrive at the quantization condition expressed in terms of periodic orbits. The periodic-orbit contribution to the trace formula can be written as the logarithmic derivative of a so-called zeta function,... 

Berry and Tabor [13] have derived a trace formula for the level density of integrable systems,... 

The periodic orbits (2.31) are referred to as bulk periodic orbits in the sense that all the F actions are nonvanishing. Therefore, all the F degrees of freedom are excited in this periodic motion. On the other hand, there exist edge periodic orbits in the subsystems in which one or several action variables vanish (see Fig. 1). These subsystems have a lower number of excited degrees of freedom, but their periodic orbits also contribute to the trace formula. However, they have smaller amplitudes, related to the amplitude of the bulk periodic orbits as... 

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. 

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... 

The experimental vibrogram shows an important recurrence around 160 fs, which may be assigned to the edge periodic orbit (3,2°, -)n0rmai- Recently, the vibrogram analysis has been carried out by Michaille et al. [113] on the basis of another model proposed by Joyeux [118] as well as on an ab initio potential fitted to the experimental data of Pique [119]. Essentially the same classical periodic orbits appear in the different models at low energies. In the same context, let us add that Joyeux has recently applied the Berry-Tabor trace formula to a IF Fermi-resonance Hamiltonian model of CS2 [120] and carried out a classical analysis of several related resonance Hamiltonians [121]. 

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]. 

Noting that TrPv = 1 according to Eq. (61) and that Lv is not a matrix but a multiplicative factor, we find that the first term in Eq. (66) is reduced to 2LV, which is independent of the order of Sv in the definition of SSim- The second term was already shown, below Eq. (52), to be equal to 2hdSb/dE. Thus, Eq. (66) recovers the trace formula (65). 

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