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Spectral Oscillations, Periodic Orbits, and Scaling

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S. A. Fulling

Quantum Physics · Submitted 14 Dec 2000

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

The authors' main claim When these matters are properly understood, the periodic-orbit theory for generic quantum systems recovers the clarity and simplicity that it always had for the wave equation in a cavity.

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arXiv IDquant-ph/0012070 (version 3)
Pith IDMEQDTAHA, Pith's permanent record of this paper and its review
CitationS. A. Fulling. "Spectral Oscillations, Periodic Orbits, and Scaling." arXiv:quant-ph/0012070 (2000). https://pith.science/paper/MEQDTAHA
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,384 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.38 · human participants 0.01 · controlled comparison 0.07 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.50 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; .

The eigenvalue density of a quantum-mechanical system exhibits oscillations, determined by the closed orbits of the corresponding classical system; this relationship is simple and strong for waves in billiards or on manifolds, but becomes slightly muddy for a Schrodinger equation with a potential, where the orbits depend on the energy. We discuss several variants of a way to restore the simplicity by rescaling the coupling constant or the size of the orbit or both. In each case the relation between the oscillation frequency and the period of the orbit is inspected critically; in many cases it is observed that a characteristic length of the orbit is a better indicator. When these matters are properly understood, the periodic-orbit theory for generic quantum systems recovers the clarity and simplicity that it always had for the wave equation in a cavity. Finally, we comment on the alleged "paradox" that semiclassical periodic-orbit theory is more effective in calculating low energy levels than high ones.

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 The eigenvalue density of a quantum-mechanical system exhibits oscillations, determined by the closed orbits of the corresponding classical system; this relationship is simple and strong for waves in billiards or on manifolds, but becomes slightly muddy for a Schrodinger equation with a potential, where the orbits depend on the energy. background 1.00 0.05 0.07 asserted 1.00
s1 We discuss several variants of a way to restore the simplicity by rescaling the coupling constant or the size of the orbit or both. contribution 0.88 0.65 0.07 asserted 0.60
s2 In each case the relation between the oscillation frequency and the period of the orbit is inspected critically; in many cases it is observed that a characteristic length of the orbit is a better indicator. result 0.84 0.84 0.13 asserted 0.54
s3 When these matters are properly understood, the periodic-orbit theory for generic quantum systems recovers the clarity and simplicity that it always had for the wave equation in a cavity. interpretation 0.95 0.34 0.79 asserted 0.99
s4 Finally, we comment on the alleged "paradox" that semiclassical periodic-orbit theory is more effective in calculating low energy levels than high ones. interpretation 0.64 0.25 0.06 unclear 0.62

What kind of paper the classifier reports this is: position or commentary 0.65 · evidence from argument 0.93.

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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

The eigenvalue density of a quantum-mechanical system exhibits oscillations, determined by the closed orbits of the corresponding classical system; this relationship is simple and strong for waves in billiards or on manifolds, but becomes slightly muddy for a Schrodinger equation with a potential, where the orbits depend on the energy. We discuss several variants of a way to restore the simplicity by rescaling the coupling constant or the size of the orbit or both. In each case the relation between the oscillation frequency and the period of the orbit is inspected critically; in many cases it is observed that a characteristic length of the orbit is a better indicator. When these matters are properly understood, the periodic-orbit theory for generic quantum systems recovers the clarity and simplicity that it always had for the wave equation in a cavity. Finally, we comment on the alleged "paradox" that semiclassical periodic-orbit theory is more effective in calculating low energy levels than high ones.

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