dorsal/arxiv
View SchemaHolonomic control operators in quantum completely integrable Hamiltonian systems
| Authors | G. Sardanashvily |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0201050 |
| URL | https://arxiv.org/abs/quant-ph/0201050 |
Abstract
We provide geometric quantization of a completely integrable Hamiltonian system in the action-angle variables around an invariant torus with respect to the angle polarization. The carrier space of this quantization is the pre-Hilbert space of smooth complex functions on the torus. A Hamiltonian of a completely integrable system in this carrier space has a countable spectrum. If it is degenerate, its eigenvalues are countably degenerate. We study nonadiabatic perturbations of this Hamiltonian by a term depending on classical time-dependent parameters. It is originated by a connection on the parameter space, and is linear in the temporal derivatives of parameters. One can choose it commuting with a degenerate Hamiltonian of a completely integrable system. Then the corresponding evolution operator acts in the eigenspaces of this Hamiltonian, and is an operator of parallel displacement along a curve in the parameter space.
{
"annotation_id": "e7cc12ae-1584-4477-8d71-53acb25d1aee",
"date_created": "2026-03-02T18:01:49.223000Z",
"date_modified": "2026-03-02T18:01:49.223000Z",
"file_hash": "7f1bb0ac069f74e81aae5ec0b3e5ca60e54d89c54872eaa9d0edad954ee5d083",
"private": false,
"record": {
"abstract": "We provide geometric quantization of a completely integrable Hamiltonian\nsystem in the action-angle variables around an invariant torus with respect to\nthe angle polarization. The carrier space of this quantization is the\npre-Hilbert space of smooth complex functions on the torus. A Hamiltonian of a\ncompletely integrable system in this carrier space has a countable spectrum. If\nit is degenerate, its eigenvalues are countably degenerate. We study\nnonadiabatic perturbations of this Hamiltonian by a term depending on classical\ntime-dependent parameters. It is originated by a connection on the parameter\nspace, and is linear in the temporal derivatives of parameters. One can choose\nit commuting with a degenerate Hamiltonian of a completely integrable system.\nThen the corresponding evolution operator acts in the eigenspaces of this\nHamiltonian, and is an operator of parallel displacement along a curve in the\nparameter space.",
"arxiv_id": "quant-ph/0201050",
"authors": [
"G. Sardanashvily"
],
"categories": [
"quant-ph",
"math-ph",
"math.MP"
],
"title": "Holonomic control operators in quantum completely integrable Hamiltonian systems",
"url": "https://arxiv.org/abs/quant-ph/0201050"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "eb022475-3388-4339-b85c-130514dab6a6",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}