dorsal/arxiv
View SchemaSpectral Oscillations, Periodic Orbits, and Scaling
| Authors | S. A. Fulling |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0012070 |
| URL | https://arxiv.org/abs/quant-ph/0012070 |
| DOI | 10.1088/0305-4470/35/18/305 |
| Journal | J. Phys. A 35, 4049-4066 (2002) |
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.
{
"annotation_id": "cd7ed5f0-8a48-40a1-8d77-9b82edb042d4",
"date_created": "2026-03-02T18:01:42.244000Z",
"date_modified": "2026-03-02T18:01:42.244000Z",
"file_hash": "5f86fed791b27c1d58704516bb849a17c3f79138a4ad37d6f608749d83ef87bd",
"private": false,
"record": {
"abstract": "The eigenvalue density of a quantum-mechanical system exhibits oscillations,\ndetermined by the closed orbits of the corresponding classical system; this\nrelationship is simple and strong for waves in billiards or on manifolds, but\nbecomes slightly muddy for a Schrodinger equation with a potential, where the\norbits depend on the energy. We discuss several variants of a way to restore\nthe simplicity by rescaling the coupling constant or the size of the orbit or\nboth. In each case the relation between the oscillation frequency and the\nperiod of the orbit is inspected critically; in many cases it is observed that\na characteristic length of the orbit is a better indicator. When these matters\nare properly understood, the periodic-orbit theory for generic quantum systems\nrecovers the clarity and simplicity that it always had for the wave equation in\na cavity. Finally, we comment on the alleged \"paradox\" that semiclassical\nperiodic-orbit theory is more effective in calculating low energy levels than\nhigh ones.",
"arxiv_id": "quant-ph/0012070",
"authors": [
"S. A. Fulling"
],
"categories": [
"quant-ph",
"nlin.CD"
],
"doi": "10.1088/0305-4470/35/18/305",
"journal_ref": "J. Phys. A 35, 4049-4066 (2002)",
"title": "Spectral Oscillations, Periodic Orbits, and Scaling",
"url": "https://arxiv.org/abs/quant-ph/0012070"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "feacfc08-476e-4b07-9ac9-cc5b2ef3522e",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}