dorsal/arxiv
View SchemaSymmetry Breaking and Bifurcations in the Periodic Orbit Theory: I: Elliptic Billiard
| Authors | A. G. Magner, S. N. Fedotkin, K. Arita, T. Misu, K. Matsuyanagi, T. Shachner, M. Brack |
|---|---|
| Categories | |
| ArXiv ID | nucl-th/9906023 |
| URL | https://arxiv.org/abs/nucl-th/9906023 |
| DOI | 10.1143/PTP.102.551 |
| Journal | Prog.Theor.Phys. 102 (1999) 551-598 |
Abstract
We derive an analytical trace formula for the level density of the two-dimensional elliptic billiard using an improved stationary phase method. The result is a continuous function of the deformation parameter (eccentricity) through all bifurcation points of the short diameter orbit and its repetitions, and possesses the correct limit of the circular billiard at zero eccentricity. Away from the circular limit and the bifurcations, it reduces to the usual (extended) Gutzwiller trace formula which for the leading-order families of periodic orbits is identical to the result of Berry and Tabor. We show that the circular disk limit of the diameter-orbit contribution is also reached through contributions from closed (periodic and non-periodic) orbits of hyperbolic type with an even number of reflections from the boundary. We obtain the Maslov indices depending on deformation and energy in terms of the phases of the complex error and Airy functions. We find enhancement of the amplitudes near the common bifurcation points of both short-diameter and hyperbolic orbits. The calculated semiclassical level densities and shell energies are in good agreement with the quantum mechanical ones.
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"abstract": "We derive an analytical trace formula for the level density of the\ntwo-dimensional elliptic billiard using an improved stationary phase method.\nThe result is a continuous function of the deformation parameter (eccentricity)\nthrough all bifurcation points of the short diameter orbit and its repetitions,\nand possesses the correct limit of the circular billiard at zero eccentricity.\nAway from the circular limit and the bifurcations, it reduces to the usual\n(extended) Gutzwiller trace formula which for the leading-order families of\nperiodic orbits is identical to the result of Berry and Tabor. We show that the\ncircular disk limit of the diameter-orbit contribution is also reached through\ncontributions from closed (periodic and non-periodic) orbits of hyperbolic type\nwith an even number of reflections from the boundary. We obtain the Maslov\nindices depending on deformation and energy in terms of the phases of the\ncomplex error and Airy functions. We find enhancement of the amplitudes near\nthe common bifurcation points of both short-diameter and hyperbolic orbits. The\ncalculated semiclassical level densities and shell energies are in good\nagreement with the quantum mechanical ones.",
"arxiv_id": "nucl-th/9906023",
"authors": [
"A. G. Magner",
"S. N. Fedotkin",
"K. Arita",
"T. Misu",
"K. Matsuyanagi",
"T. Shachner",
"M. Brack"
],
"categories": [
"nucl-th",
"chao-dyn",
"nlin.CD"
],
"doi": "10.1143/PTP.102.551",
"journal_ref": "Prog.Theor.Phys. 102 (1999) 551-598",
"title": "Symmetry Breaking and Bifurcations in the Periodic Orbit Theory: I: Elliptic Billiard",
"url": "https://arxiv.org/abs/nucl-th/9906023"
},
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