## Semiclassical Transition from an Elliptical to an Oval (1997)

Venue: | Billiard, J. Phys. A |

Citations: | 7 - 3 self |

### BibTeX

@ARTICLE{Sieber97semiclassicaltransition,

author = {Martin Sieber and Abteilung Theoretische Physik},

title = {Semiclassical Transition from an Elliptical to an Oval},

journal = {Billiard, J. Phys. A},

year = {1997},

pages = {4563--4596}

}

### Years of Citing Articles

### OpenURL

### Abstract

Semiclassical approximations often involve the use of stationary phase approximations. This method can be applied when ¯h is small in comparison to relevant actions or action differences in the corresponding classical system. In many situations, however, action differences can be arbitrarily small and then uniform approximations are more appropriate. In the present paper we examine different uniform approximations for describing the spectra of integrable systems and systems with mixed phase space. This is done on the example of two billiard systems, an elliptical billiard and a deformation of it, an oval billiard. We derive a trace formula for the ellipse which involves a uniform approximation for the Maslov phases near the separatrix, and a uniform approximation for tori of periodic orbits close to a bifurcation. We then examine how the trace formula is modified when the ellipse is deformed into an oval. This involves uniform approximations for the break-up of tori and uniform approximations for bifurcations of periodic orbits. Relations between different uniform approximations are discussed. PACS numbers: 03.65.Ge Solutions of wave equations: bound states. 03.65.Sq Semiclassical theories and applications. 05.45.+b Theory and models of chaotic systems.

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Citation Context ...occur, most of them for long periodic orbits. Bifurcations occur in different forms, but the number of generic forms in two-dimensional systems is finite. They were classified by Meyer [11] and Bruno =-=[12, 13]-=-. Their form depends on the repetition number m of an orbit that bifurcates. The semiclassical treatment of these generic bifurcations was investigated by Ozorio de Almeida and Hannay [14]. They deriv... |

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Citation Context ...riodic orbits near bifurcations in terms of standard diffraction catastrophe integrals. These approximations have for example been applied for treating tangent bifurcations and pitchfork bifurcations =-=[15, 16]-=-. For the largest class of bifurcations with m > 4 the results of Ozorio de Almeida and Hannay were extended in [17] by including higher order terms in the normal form expansion for the bifurcation. I... |

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Citation Context ... number of periodic orbits which have small action differences when the perturbation is small. Ozorio de Almeida derived a uniform approximation which is valid if the splitting of the orbits is small =-=[18]-=-. A formula for the break-up of families of orbits due to more general symmetries was derived by Creagh [19]. Tomsovic, Grinberg and Ullmo extended the result of Ozorio de Almeida and obtained a unifo... |

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Citation Context ...eida derived a uniform approximation which is valid if the splitting of the orbits is small [18]. A formula for the break-up of families of orbits due to more general symmetries was derived by Creagh =-=[19]-=-. Tomsovic, Grinberg and Ullmo extended the result of Ozorio de Almeida and obtained a uniform approximation which interpolates between the torus contribution and Gutzwiller’s approximation [20, 21]. ... |

1 |
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Citation Context ... investigated under different view points. This includes the treatment of the action-angle variables [26], the caustics of the classical motion [26, 27], Poncelet’s theorem [28, 29], the billiard map =-=[30, 31]-=-, and the periodic orbits [29, 32, 33]. A treatment of the Schrödinger equation for the elliptical billiard can be found in [34]. In the following we briefly review classical and quantum properties of... |

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Citation Context .... e. to a product of trigonometric and Bessel functions [34]. A numerical examination of the energy spectrum of the elliptical billiard in dependence on the ratio of the two half-axis can be found in =-=[36]-=-. 2.3 The semiclassical approximation for the elliptical billiard In this section we derive a semiclassical trace formula for the spectral density of the elliptical billiard. We follow the method of B... |

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Citation Context ...y phase approximations which connect both approximations are all only valid in leading order of ¯h. A similar result was observed previously for a circular billiard with a singular magnetic flux line =-=[41]-=-. The smallness of D EBK(x) − D SC(x) shows 19also that the modifications for contributions of tori near the separatrix due to Eq. (46) are not large in the range where the numerical examination was ... |