Periodic-orbit theory

In fig. 26 the Arrhenius plot ln[k(r)/coo] versus TojT = Pl2n is shown for V /(Oo = 3, co = 0.1, C = 0.0357. The disconnected points are the data from Hontscha et al. [1990]. The solid line was obtained with the two-dimensional instanton method. One sees that the agreement between the instanton result and the exact quantal calculations is perfect. The low-temperature limit found with the use of the periodic-orbit theory expression for kio (dashed line) also excellently agrees with the exact result. Figure 27 presents the dependence ln(/Cc/( o) on the coupling strength defined as C fQ. The dashed line corresponds to the exact result from Hontscha et al. [1990], and the disconnected points are obtained with the instanton method. For most practical purposes the instanton results may be considered exact. 

Here we shall describe how the periodic-orbit theory of section 3.4, relating the energy levels with the poles of the spectral function g E), can be extended to two dimensions. For simplicity we shall exemplify this extension by the simplest model in which the total PES is constructed of two paraboloids crossing at some dividing line. Each paraboloid is characterized by two eigenfrequen-cies, o + and [Pg.72]

In accordance with the one-dimensional periodic orbit theory, any orbit contributing to g E) is supposedly constructed from closed classical orbits in the well and subbarrier imaginary-time trajectories. These two classes of trajectories are bordering on the turning points. For the present model the classical motion in the well is separable, and the harmonic approximation for classical motion is quite reasonable for more realistic potentials, if only relatively low energy levels are involved. 

J. Kaidel, P. Winkler, and M. Brack, Periodic orbit theory for the Henon-Heiles system in the continuum region, Phys. Rev. E 70, 066208 (2004). 

B. Eckhardt, in Periodic Orbit Theory, Proceedings of the International School of Physics, Course CXIX, G. Casati, I. Guarneri, and U. Smilansky, eds., North-Holland, Amsterdam, 1993. 

Certain semiclassical properties involving the eigenfunctions can also be calculated with periodic-orbit theory. Considering the Wigner functions corresponding to die energy eigenfunctions H = Enn [28],... 

Periodic-orbit theory provides the unique semiclassical quantization scheme for nonseparable systems with a fully chaotic and fractal iepeller. As we mentioned in Section II, the different periodic orbits of the repeller have quantum amplitudes weighted by the stability eigenvalues, and the periodic-orbit amplitudes interfere among each other as described by the zeta function. The more unstable the periodic orbit is, the less it contributes in (2.24). Therefore, only the least unstable periodic orbits play a dominant role. 

I. Burhardt, P. Gaspard, Molecular transition state, resonances, and periodic orbit theory, J. Chem. Phys. 100 (1994) 6395. 

The purpose of this chapter is to review some properties of isomerizing (ABC BCA) and dissociating (ABC AB + C) prototype triatomic molecules, which are revealed by the analysis of their dynamics on precise ab initio potential energy surfaces (PESs). The systems investigated will be considered from all possible viewpoints—quanmm, classical, and semiclassical mechanics—and several techniques will be applied to extract information from the PES, such as Canonical Perturbation Theory, adiabatic separation of motions, and Periodic Orbit Theory. 

The most recent advance in the theory of the helium atom was the discovery of its classically chaotic nature. In connection with modern semiclassical techniques, such as Gutzwiller s periodic orbit theory and cycle expansion techniques, it was possible to obtain substantial new insight into the structure of doubly excited states of two-electron atoms and ions. This new direction in the application of chaos in atomic physics was initiated by Ezra et al. (1991), Kim and Ezra (1991), Richter (1991), and Bliimel and Reinhardt (1992). The discussion of the manifestations of chaos in the helium atom is the focus of this chapter. 

Novel resonance effects occur in nonhydrogenic atoms in a strong magnetic field. As noted by Dando et al. (1995) these effects cannot be explained on the basis of hydrogenic periodic orbit theory. Their relation to ghost orbits is discussed by Hiipper et al. (1995). 

Vattay, G., Wirzba, A. and Rosenqvist, P.E. (1994). Periodic orbit theory of diffraction, Phys. Rev. Lett. 73, 2304-2307. 

ABSTRACT. The study of periodic orbits embedded in the continuum has provided a new tool for understanding the dynamics of molecular collisions, The application of periodic orbit theory to classical variational transition state theory, quantal threshold and resonance effects is presented. Special emphasis is given to the stability analysis of periodic orbits in collinear and three dimensional systems. Future applications of periodic orbit theory are outlined. 

In Section 2 of this paper a brief account of the 3D q -HO is given, while in Section 3 the appUcation of the periodic orbit theory of Balian and Bloch to metal clusters is briefly described. The predictions of the 3D g-HO model axe compared to the restrictions imposed by the theory of Balian and Bloch in Section 4, while in Section 5 a modified Hamiltonian for the 3D q-HO is introduced, allowing for full agreement with the theory of Balian and... 

In this paper we have attempted a comparison of the predictions for the shell structure of metal clusters of the 3D g-HO model to the ones of the periodic orbit theory of Balian and Bloch. It turns out that the predictions of the 3D g-HO model for the magic numbers of metal clusters can be... 

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