dorsal/arxiv
View SchemaPure quantum integrability
| Authors | Jarmo Hietarinta |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9708010 |
| URL | https://arxiv.org/abs/solv-int/9708010 |
| DOI | 10.1016/S0375-9601(98)00535-0 |
Abstract
The correspondence between the integrability of classical mechanical systems and their quantum counterparts is not a 1-1, although some close correspondencies exist. If a classical mechanical system is integrable with invariants that are polynomial in momenta one can construct a corresponding commuting set of differential operators. Here we discuss some 2- or 3-dimensional purely quantum integrable systems (the 1-dimensional counterpart is the Lame equation). That is, we have an integrable potential whose amplitude is not free but rather proportional to $\hbar^2$, and in the classical limit the potential vanishes. Furthermore it turns out that some of these systems actually have N+1 commuting differential operators, connected by a nontrivial algebraic relation. Some of them have been discussed recently by A.P. Veselov et. al.} from the point of view of Baker-Akheizer functions.
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"abstract": "The correspondence between the integrability of classical mechanical systems\nand their quantum counterparts is not a 1-1, although some close\ncorrespondencies exist. If a classical mechanical system is integrable with\ninvariants that are polynomial in momenta one can construct a corresponding\ncommuting set of differential operators. Here we discuss some 2- or\n3-dimensional purely quantum integrable systems (the 1-dimensional counterpart\nis the Lame equation). That is, we have an integrable potential whose amplitude\nis not free but rather proportional to $\\hbar^2$, and in the classical limit\nthe potential vanishes. Furthermore it turns out that some of these systems\nactually have N+1 commuting differential operators, connected by a nontrivial\nalgebraic relation. Some of them have been discussed recently by A.P. Veselov\net. al.} from the point of view of Baker-Akheizer functions.",
"arxiv_id": "solv-int/9708010",
"authors": [
"Jarmo Hietarinta"
],
"categories": [
"solv-int",
"nlin.SI"
],
"doi": "10.1016/S0375-9601(98)00535-0",
"title": "Pure quantum integrability",
"url": "https://arxiv.org/abs/solv-int/9708010"
},
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