dorsal/arxiv
View SchemaOrdered Products, $W_{\infty}$-Algebra, and Two-Variable, Definite-Parity, Orthogonal Polynomials
| Authors | A. Verçin |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9712026 |
| URL | https://arxiv.org/abs/quant-ph/9712026 |
| DOI | 10.1063/1.532295 |
| Journal | J.Math.Phys. 39 (1998) 2418-2427 |
Abstract
It has been shown that the Cartan subalgebra of $W_{\infty}$- algebra is the space of the two-variable, definite-parity polynomials. Explicit expressions of these polynomials, and their basic properties are presented. Also has been shown that they carry the infinite dimensional irreducible representation of the $su(1,1)$ algebra having the spectrum bounded from below. A realization of this algebra in terms of difference operators is also obtained. For particular values of the ordering parameter $s$ they are identified with the classical orthogonal polynomials of a discrete variable, such as the Meixner, Meixner-Pollaczek, and Askey-Wilson polynomials. With respect to variable $s$ they satisfy a second order eigenvalue equation of hypergeometric type. Exact scattering states with zero energy for a family of potentials are expressed in terms of these polynomials. It has been put forward that it is the \.{I}n\"{o}n\"{u}-Wigner contraction and its inverse that form bridge between the difference and differential calculus.
{
"annotation_id": "d2f539af-566e-4d17-bfb2-68061c985d28",
"date_created": "2026-03-02T18:02:41.047000Z",
"date_modified": "2026-03-02T18:02:41.047000Z",
"file_hash": "be120e25c51eb4da615a0ac29df5bd3869781675b005e0427159878236b8f09a",
"private": false,
"record": {
"abstract": "It has been shown that the Cartan subalgebra of $W_{\\infty}$- algebra is the\nspace of the two-variable, definite-parity polynomials. Explicit expressions of\nthese polynomials, and their basic properties are presented. Also has been\nshown that they carry the infinite dimensional irreducible representation of\nthe $su(1,1)$ algebra having the spectrum bounded from below. A realization of\nthis algebra in terms of difference operators is also obtained. For particular\nvalues of the ordering parameter $s$ they are identified with the classical\northogonal polynomials of a discrete variable, such as the Meixner,\nMeixner-Pollaczek, and Askey-Wilson polynomials. With respect to variable $s$\nthey satisfy a second order eigenvalue equation of hypergeometric type. Exact\nscattering states with zero energy for a family of potentials are expressed in\nterms of these polynomials. It has been put forward that it is the\n\\.{I}n\\\"{o}n\\\"{u}-Wigner contraction and its inverse that form bridge between\nthe difference and differential calculus.",
"arxiv_id": "quant-ph/9712026",
"authors": [
"A. Ver\u00e7in"
],
"categories": [
"quant-ph"
],
"doi": "10.1063/1.532295",
"journal_ref": "J.Math.Phys. 39 (1998) 2418-2427",
"title": "Ordered Products, $W_{\\infty}$-Algebra, and Two-Variable, Definite-Parity, Orthogonal Polynomials",
"url": "https://arxiv.org/abs/quant-ph/9712026"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "a6349a44-6951-4aaf-ac08-2d195c14e6c8",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}