Gutzwiller

Gutzwiller M C 1990 Chaos in Classical and Quantum Mechanics (New York Springer)... 

Gutzwiller M C 1967 Phase-integral approximation in momentum space and the bound states of an atom J Math. Phys. 8 1979... 

This equation shows that only the closed classical trajectories [x(t) = x(0) and x(t) = x(0)] should be taken into account, and the energy spectrum is determined by these periodic orbits [Gutzwiller 1967 Balian and Bloch 1974 Miller 1975a Rajaraman 1975]. 

It is noteworthy that eq. (4.15a) is nothing but the linearized classical upside-down barrier equation of motion (8S/8x = 0) for the new coordinate x. Therefore, while x = 0 corresponds to the instanton, the nonzero solution to (4.15a) describes how the trajectory escapes from the instanton solution, when it deviates from it. The parameter X, referred to as the stability angle [Gutzwil-ler 1967 Rajaraman 1975], generalizes the harmonic-oscillator phase co, which would appear in (4.15), if CO, were a constant. The fact that X is real indicates the aforementioned instability of the instanton in two dimensions. Guessing that the determinant det( — -I- co, ) is a function of X only,... 

Steffen T, Christen S, Blattler R and Gutzwiller F (2001). Infectious diseases and public health Risk taking behaviour during participation in the Swiss Programme for a Medical Prescription of Narcotics (PROVE). Journal of Substance Use and Misuse, 36, 71-89. 

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,... 

In this chapter we present some results obtained by our group on scar theory in the context of molecular vibrations, and in particular for the LiNC/LiCN molecular system. This kind of (generic) systems exhibits a dynamical behavior in which regular and chaotic motions are mixed (Gutzwiller, 1990), a situation which presents significant differences with respect to the completely chaotic case considered in most references cited above, and are very important in many areas of physics and chemistry. 

Gutzwiller, M. C. Chaos in Classical and Quantum Mechanics Springer-Verlag, New York, 1990. 

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. 

One of the main characteristics of the statistical properies of the spectra is the level spacing distribution (Eckhardt, 1988 Gutzwiller, 1990) function. In this work we calculate the nearest-neighbor levelspacing distribution (Eckhardt, 1988 Gutzwiller, 1990). 

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

The evaluation of the semiclassical Van Vleck-Gutzwiller propagator (106) amounts to the solution of a boundary-value problem. That is, given a trajectory characterized by the position q(f) = q, and momentum p(f) = p, we need to hnd the roots of the equation = q floiPo)- To circumvent this cumbersome root search, one may rewrite the semiclassical expression for the transition amplitude (105) as an initial-value problem [104-111]... 

In contrast to the quasi-classical approaches discussed in the previous chapters of this review, Eq. (114) represents a description of nonadiabatic dynamics which is semiclassically exact in the sense that it requires only the basic semiclassical Van Vleck-Gutzwiller approximation [3] to the quantum propagator. Therefore, it allows the description of electronic and nuclear quantum effects. 

---