Semiclassical Quantization Using Diffractive Orbits

Whelan, Niall D.

doi:10.1103/physrevlett.76.2605

Cited by 16 publications

(18 citation statements)

References 29 publications

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“…Using the GTD approximation the contributions of diffracting orbits were calculated in a number of papers [53,54,55]. These contributions as well as the Berry-Tabor trace formula for the rectangular billiard are derived in this Appendix, for completeness.…”

Section: Appendix A: Diffracting Orbits Contributionsmentioning

confidence: 99%

Spectral statistics of rectangular billiards with localized perturbations

Rahav

Fishman

2002

Nonlinearity

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The form factor K(τ ) is calculated analytically to the order τ 3 as well as numerically for a rectangular billiard perturbed by a δ-like scatterer with an angle independent diffraction constant, D. The cases where the scatterer is at the center and at a typical position in the billiard are studied. The analytical calculations are performed in the semiclassical approximation combined with the geometrical theory of diffraction. Non diagonal contributions are crucial and are therefore taken into account. The numerical calculations are performed for a self adjoint extension of a δ function potential. We calculate the angle dependent diffraction constant for an arbitrary perturbing potential U (r), that is large in a finite but small region (compared to the wavelength of the particles that in turn is small compared to the size of the billiard). The relation to the idealized model of the δ-like scatterer is formulated. The angle dependent diffraction constant is used for the analytic calculation of the form factor to the order τ 2 . If the scatterer is at a typical position, the form factor is found to reduce (in this order) to the one found for angle independent diffraction. If the scatterer is at the center, the large degeneracy in the lengths of the orbits involved leads to an additional small contribution to the form factor, resulting of the angle dependence of the diffraction constant. The robustness of the results is discussed.

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Section: Appendix A: Diffracting Orbits Contributionsmentioning

confidence: 99%

Spectral statistics of rectangular billiards with localized perturbations

Rahav

Fishman

2002

Nonlinearity

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“…The cardioid billiard also has re ection symmetry with respect to the x-axis X : (s; p) 7 ! ( s; p): (31) X and T are involutions, i.e. T 2 = id, X 2 = id and det T = 1, det X = 1.…”

Section: Symmetriesmentioning

confidence: 99%

“…In the quantum mechanical billiard problem discontinuities of derivatives of the boundary curve can play an important role (see, e.g. 25,26,27] and 28,29,30,12,31] and references therein). The cusp orbits start at an arbitrary angle in the cusp and eventually return to the cusp with an arbitrary angle without the need to ful ll any re ection condition in the cusp.…”

Section: Finite Orbitsmentioning

confidence: 99%

Symbolic dynamics and periodic orbits for the cardioid billiard

Bäcker

Dullin

1997

J. Phys. A: Math. Gen.

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Abstract:The periodic orbits of the strongly chaotic cardioid billiard are studied by introducing a binary symbolic dynamics. The corresponding partition is mapped to a topological well-ordered symbol plane. In the symbol plane the pruning front is obtained from orbits running either into or through the cusp. We show that all periodic orbits correspond to maxima of the Lagrangian and give a complete list up to code length 15. The symmetry reduction is done on the level of the symbol sequences and the periodic orbits are classi ed using symmetry lines. We show that there exists an in nite number of families of periodic orbits accumulating in length and that all other families of geometrically short periodic orbits eventually get pruned. All these orbits are related to nite orbits starting and ending in the cusp. We obtain an analytical estimate of the Kolmogorov-Sinai entropy and nd good agreement with the numerically calculated value and the one obtained by averaging periodic orbits. Furthermore the statistical properties of periodic orbits are investigated.

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“…This was also noted in the scattering geometries discussed in Refs. [7,9] where it caused there to be no lower families of quantum resonances. This difference is intimately related to the fact that the Green functions in Eq.…”

Section: Trace Formula For Diffractive Orbitsmentioning

confidence: 99%

Edge diffraction, trace formulae and the cardioid billiard

Bruus

Whelan²

1996

Nonlinearity

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We study the effect of edge diffraction on the semiclassical analysis of two dimensional quantum systems by deriving a trace formula which incorporates paths hitting any number of vertices embedded in an arbitrary potential. This formula is used to study the cardioid billiard, which has a single vertex. The formula works well for most of the short orbits we analyzed but fails for a few diffractive orbits due to a breakdown in the formalism for certain geometries. We extend the symbolic dynamics to account for diffractive orbits and use it to show that in the presence of parity symmetry the trace formula decomposes in an elegant manner such that for the cardioid billiard the diffractive orbits have no effect on the odd spectrum. Including diffractive orbits helps resolve peaks in the density of even states but does not appear to affect their positions. An analysis of the level statistics shows no significant difference between spectra with and without diffraction.

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Semiclassical Quantization Using Diffractive Orbits

Cited by 16 publications

References 29 publications

Spectral statistics of rectangular billiards with localized perturbations

Spectral statistics of rectangular billiards with localized perturbations

Symbolic dynamics and periodic orbits for the cardioid billiard

Edge diffraction, trace formulae and the cardioid billiard

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