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How Chaotic is the Stadium Billiard? A Semiclassical Analysis
chao-dyn · Submitted 14 Oct 1996
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim I show that this belief is erroneous.
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The abstract, sentence by sentence
The impression gained from the literature published to date is that the spectrum of the stadium billiard can be adequately described, semiclassically, by the Gutzwiller periodic orbit trace formula together with a modified treatment of the marginally stable family of bouncing ball orbits. I show that this belief is erroneous. The Gutzwiller trace formula is not applicable for the phase space dynamics near the bouncing ball orbits. Unstable periodic orbits close to the marginally stable family in phase space cannot be treated as isolated stationary phase points when approximating the trace of the Green function. Semiclassical contributions to the trace show an $\hbar$ - dependent transition from hard chaos to integrable behavior for trajectories approaching the bouncing ball orbits. A whole region in phase space surrounding the marginal stable family acts, semiclassically, like a stable island with boundaries being explicitly $\hbar$-dependent. The localized bouncing ball states found in the billiard derive from this semiclassically stable island. The bouncing ball orbits themselves, however, do not contribute to individual eigenvalues in the spectrum. An EBK-like quantization of the regular bouncing ball eigenstates in the stadium can be derived. The stadium billiard is thus an ideal model for studying the influence of almost regular dynamics near marginally stable boundaries on quantum mechanics.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The impression gained from the literature published to date is that the spectrum of the stadium billiard can be adequately described, semiclassically, by the Gutzwiller periodic orbit trace formula together with a modified treatment of the marginally stable family of bouncing ball orbits. | background 1.00 | 0.03 | 0.04 | hedged possibility 0.70 |
| s1 | I show that this belief is erroneous. | result 0.39 | 0.61 | 0.33 | asserted 1.00 |
| s2 | The Gutzwiller trace formula is not applicable for the phase space dynamics near the bouncing ball orbits. | result 0.99 | 0.71 | 0.41 | asserted 1.00 |
| s3 | Unstable periodic orbits close to the marginally stable family in phase space cannot be treated as isolated stationary phase points when approximating the trace of the Green function. | result 0.96 | 0.74 | 0.22 | asserted 1.00 |
| s4 | Semiclassical contributions to the trace show an $\hbar$ - dependent transition from hard chaos to integrable behavior for trajectories approaching the bouncing ball orbits. | result 1.00 | 0.85 | 0.21 | asserted 1.00 |
| s5 | A whole region in phase space surrounding the marginal stable family acts, semiclassically, like a stable island with boundaries being explicitly $\hbar$-dependent. | result 0.97 | 0.85 | 0.29 | asserted 1.00 |
| s6 | The localized bouncing ball states found in the billiard derive from this semiclassically stable island. | result 0.76 | 0.81 | 0.12 | asserted 1.00 |
| s7 | The bouncing ball orbits themselves, however, do not contribute to individual eigenvalues in the spectrum. | result 1.00 | 0.82 | 0.12 | asserted 1.00 |
| s8 | An EBK-like quantization of the regular bouncing ball eigenstates in the stadium can be derived. | result 0.79 | 0.88 | 0.07 | asserted 0.97 |
| s9 | The stadium billiard is thus an ideal model for studying the influence of almost regular dynamics near marginally stable boundaries on quantum mechanics. | interpretation 0.97 | 0.33 | 0.13 | asserted 1.00 |
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Original abstract
The impression gained from the literature published to date is that the spectrum of the stadium billiard can be adequately described, semiclassically, by the Gutzwiller periodic orbit trace formula together with a modified treatment of the marginally stable family of bouncing ball orbits. I show that this belief is erroneous. The Gutzwiller trace formula is not applicable for the phase space dynamics near the bouncing ball orbits. Unstable periodic orbits close to the marginally stable family in phase space cannot be treated as isolated stationary phase points when approximating the trace of the Green function. Semiclassical contributions to the trace show an $\hbar$ - dependent transition from hard chaos to integrable behavior for trajectories approaching the bouncing ball orbits. A whole region in phase space surrounding the marginal stable family acts, semiclassically, like a stable island with boundaries being explicitly $\hbar$-dependent. The localized bouncing ball states found in the billiard derive from this semiclassically stable island. The bouncing ball orbits themselves, however, do not contribute to individual eigenvalues in the spectrum. An EBK-like quantization of the regular bouncing ball eigenstates in the stadium can be derived. The stadium billiard is thus an ideal model for studying the influence of almost regular dynamics near marginally stable boundaries on quantum mechanics.
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