dorsal/arxiv
View SchemaSome Exact Results for Mid-Band and Zero Band-Gap States of Associated Lame Potentials
| Authors | Avinash Khare, Uday Sukhatme |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0105044 |
| URL | https://arxiv.org/abs/quant-ph/0105044 |
| DOI | 10.1063/1.1416487 |
| Journal | J.Math.Phys. 42 (2001) 5652-5664 |
Abstract
Applying certain known theorems about one-dimensional periodic potentials, we show that the energy spectrum of the associated Lam\'{e} potentials $$a(a+1)m~{\rm sn}^2(x,m)+b(b+1)m~{\rm cn}^2(x,m)/{\rm dn}^2(x,m)$$ consists of a finite number of bound bands followed by a continuum band when both $a$ and $b$ take integer values. Further, if $a$ and $b$ are unequal integers, we show that there must exist some zero band-gap states, i.e. doubly degenerate states with the same number of nodes. More generally, in case $a$ and $b$ are not integers, but either $a + b$ or $a - b$ is an integer ($a \ne b$), we again show that several of the band-gaps vanish due to degeneracy of states with the same number of nodes. Finally, when either $a$ or $b$ is an integer and the other takes a half-integral value, we obtain exact analytic solutions for several mid-band states.
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"abstract": "Applying certain known theorems about one-dimensional periodic potentials, we\nshow that the energy spectrum of the associated Lam\\\u0027{e} potentials\n$$a(a+1)m~{\\rm sn}^2(x,m)+b(b+1)m~{\\rm cn}^2(x,m)/{\\rm dn}^2(x,m)$$ consists of\na finite number of bound bands followed by a continuum band when both $a$ and\n$b$ take integer values. Further, if $a$ and $b$ are unequal integers, we show\nthat there must exist some zero band-gap states, i.e. doubly degenerate states\nwith the same number of nodes. More generally, in case $a$ and $b$ are not\nintegers, but either $a + b$ or $a - b$ is an integer ($a \\ne b$), we again\nshow that several of the band-gaps vanish due to degeneracy of states with the\nsame number of nodes. Finally, when either $a$ or $b$ is an integer and the\nother takes a half-integral value, we obtain exact analytic solutions for\nseveral mid-band states.",
"arxiv_id": "quant-ph/0105044",
"authors": [
"Avinash Khare",
"Uday Sukhatme"
],
"categories": [
"quant-ph",
"cond-mat",
"hep-th",
"math-ph",
"math.MP"
],
"doi": "10.1063/1.1416487",
"journal_ref": "J.Math.Phys. 42 (2001) 5652-5664",
"title": "Some Exact Results for Mid-Band and Zero Band-Gap States of Associated Lame Potentials",
"url": "https://arxiv.org/abs/quant-ph/0105044"
},
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