dorsal/arxiv
View SchemaDoes a quantum particle know the time?
| Authors | Lev Kapitanski, Igor Rodnianski |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9711062 |
| URL | https://arxiv.org/abs/quant-ph/9711062 |
Abstract
We study the spatial regularity of the fundamental solution E(t,x) of the Schr\"odinger equation on the circle in a scale of Besov spaces. Although the fundamental solution is not smooth, we reveal a fine change of regularity of E(t,x) at different times t. For rational t, E(t,x) is a weighted sum of delta-functions, and, therefore, exhibits the same regularity as at t=0. For irrational t, the regularity of E(t,x) is better and depends on how well t is approximated by rationals. For badly approximated t (e.g., when t is a quadratic irrational, or, more generally, when t has bounded quotients in its continued fraction expansion), E(t,x) is a "1/2-derivative" more regular than E(0,x). For a generic irrational t, E(t,x) is almost "1/2-derivative" more regular. However, the better t is approximated by rationals, the lower is the regularity of E(t,x). We describe different thin classes of irrationals which prescribe their particular regularity to the fundamental solution. These classes are singled out and characterized by the behavior of the continued fraction expansions of their members.
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"abstract": "We study the spatial regularity of the fundamental solution E(t,x) of the\nSchr\\\"odinger equation on the circle in a scale of Besov spaces. Although the\nfundamental solution is not smooth, we reveal a fine change of regularity of\nE(t,x) at different times t. For rational t, E(t,x) is a weighted sum of\ndelta-functions, and, therefore, exhibits the same regularity as at t=0. For\nirrational t, the regularity of E(t,x) is better and depends on how well t is\napproximated by rationals. For badly approximated t (e.g., when t is a\nquadratic irrational, or, more generally, when t has bounded quotients in its\ncontinued fraction expansion), E(t,x) is a \"1/2-derivative\" more regular than\nE(0,x). For a generic irrational t, E(t,x) is almost \"1/2-derivative\" more\nregular. However, the better t is approximated by rationals, the lower is the\nregularity of E(t,x). We describe different thin classes of irrationals which\nprescribe their particular regularity to the fundamental solution. These\nclasses are singled out and characterized by the behavior of the continued\nfraction expansions of their members.",
"arxiv_id": "quant-ph/9711062",
"authors": [
"Lev Kapitanski",
"Igor Rodnianski"
],
"categories": [
"quant-ph"
],
"title": "Does a quantum particle know the time?",
"url": "https://arxiv.org/abs/quant-ph/9711062"
},
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