dorsal/arxiv
View SchemaA particle in the Bio-Savart-Laplace magnetic field: explicit solutions
| Authors | D. Yafaev |
|---|---|
| Categories | |
| ArXiv ID | physics/0403131 |
| URL | https://arxiv.org/abs/physics/0403131 |
Abstract
We consider the Schr\"odinger operator ${\bf H}=(i\nabla+A)^2 $ in the space $L_2({\mathbb R}^3)$ with a magnetic potential $A $ created by an infinite straight current. We perform a spectral analysis of the operator ${\bf H}$ almost explicitly. In particular, we show that the operator ${\bf H}$ is absolutely continuous, its spectrum has infinite multiplicity and coincides with the positive half-axis. Then we find the large-time behavior of solutions $\exp(-i{\bf H}t)f$ of the time dependent Schr\"odinger equation. Equations of classical mechanics are also integrated. Our main observation is that both quantum and classical particles have always a preferable (depending on its charge) direction of propagation along the current and both of them are confined in the plane orthogonal to the current.
{
"annotation_id": "0f932c22-a0ca-4359-9bc6-94b2a6f8b073",
"date_created": "2026-03-02T18:00:50.328000Z",
"date_modified": "2026-03-02T18:00:50.328000Z",
"file_hash": "bf897e6c56dc3cfebdd1986ddcf872d5753ea58168430778a9755e5b8ff42498",
"private": false,
"record": {
"abstract": "We consider the Schr\\\"odinger operator ${\\bf H}=(i\\nabla+A)^2 $ in the space\n$L_2({\\mathbb R}^3)$ with a magnetic potential $A $ created by an infinite\nstraight current. We perform a spectral analysis of the operator ${\\bf H}$\nalmost explicitly.\n In particular, we show that the operator ${\\bf H}$ is absolutely continuous,\nits spectrum has infinite multiplicity and coincides with the positive\nhalf-axis. Then we find the large-time behavior of solutions $\\exp(-i{\\bf\nH}t)f$ of the time dependent Schr\\\"odinger equation. Equations of classical\nmechanics are also integrated. Our main observation is that both quantum and\nclassical particles have always a preferable (depending on its charge)\ndirection of propagation along the current and both of them are confined in the\nplane orthogonal to the current.",
"arxiv_id": "physics/0403131",
"authors": [
"D. Yafaev"
],
"categories": [
"physics.plasm-ph",
"physics.gen-ph"
],
"title": "A particle in the Bio-Savart-Laplace magnetic field: explicit solutions",
"url": "https://arxiv.org/abs/physics/0403131"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "f5cef917-a619-4e00-93f3-2e04a47e07a7",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}