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Numerical accuracy of Bogomolny's semiclassical quantization scheme in quantum billiards

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Bambi Hu, Baowen Li, Daniel C Rouben (Centre for Nonlinear Studies, Department of Physics, Hong Kong Baptist University, China)

chao-dyn · Submitted 9 Jun 1999

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim The classical dynamics of this billiard system undergoes a smooth transition from integrable (circle) to completely chaotic motion, thus providing a test of Bogomolny's semiclassical method in coordinate space in terms of the morphology of the wavefunction.

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arXiv IDchao-dyn/9906013 (version 1)
Pith IDS4ZBW7P6, Pith's permanent record of this paper and its review
CitationBambi Hu, Baowen Li, China), Daniel C Rouben (Centre for Nonlinear Studies, Department of Physics, Hong Kong Baptist University. "Numerical accuracy of Bogomolny's semiclassical quantization scheme in quantum billiards." arXiv:chao-dyn/9906013 (1999). https://pith.science/paper/S4ZBW7P6
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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,374 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.16 · human participants 0.01 · controlled comparison 0.31 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.92 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We use the semiclassical quantization scheme of Bogomolny to calculate eigenvalues of the Lima\c con quantum billiard corresponding to a conformal map of the circle billiard. We use the entire billiard boundary as the chosen surface of section (SOS) and use a finite approximation for the transfer operator in coordinate space. Computation of the eigenvalues of this matrix combined with a quantization condition, determines a set of semiclassical eigenvalues which are compared with those obtained by solving the Schr\"odinger equation. The classical dynamics of this billiard system undergoes a smooth transition from integrable (circle) to completely chaotic motion, thus providing a test of Bogomolny's semiclassical method in coordinate space in terms of the morphology of the wavefunction. We analyze the results for billiards which exhibit both soft and hard chaos.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We use the semiclassical quantization scheme of Bogomolny to calculate eigenvalues of the Lima\c con quantum billiard corresponding to a conformal map of the circle billiard. contribution 0.64 0.95 0.05 asserted 0.98
s1 We use the entire billiard boundary as the chosen surface of section (SOS) and use a finite approximation for the transfer operator in coordinate space. method 1.00 0.95 0.03 asserted 0.98
s2 Computation of the eigenvalues of this matrix combined with a quantization condition, determines a set of semiclassical eigenvalues which are compared with those obtained by solving the Schr\"odinger equation. method 0.98 0.93 0.09 asserted 0.99
s3 The classical dynamics of this billiard system undergoes a smooth transition from integrable (circle) to completely chaotic motion, thus providing a test of Bogomolny's semiclassical method in coordinate space in terms of the morphology of the wavefunction. background 0.37 0.40 0.12 asserted 0.99
s4 We analyze the results for billiards which exhibit both soft and hard chaos. contribution 0.81 0.88 0.06 asserted 0.89

What kind of paper the classifier reports this is: simulation study 1.00 · evidence from simulation or computation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

We use the semiclassical quantization scheme of Bogomolny to calculate eigenvalues of the Lima\c con quantum billiard corresponding to a conformal map of the circle billiard. We use the entire billiard boundary as the chosen surface of section (SOS) and use a finite approximation for the transfer operator in coordinate space. Computation of the eigenvalues of this matrix combined with a quantization condition, determines a set of semiclassical eigenvalues which are compared with those obtained by solving the Schr\"odinger equation. The classical dynamics of this billiard system undergoes a smooth transition from integrable (circle) to completely chaotic motion, thus providing a test of Bogomolny's semiclassical method in coordinate space in terms of the morphology of the wavefunction. We analyze the results for billiards which exhibit both soft and hard chaos.

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