dorsal/arxiv
View SchemaComplex Periodic Potentials with a Finite Number of Band Gaps
| Authors | Avinash Khare, Uday Sukhatme |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0602105 |
| URL | https://arxiv.org/abs/quant-ph/0602105 |
| DOI | 10.1063/1.2204810 |
Abstract
We obtain several new results for the complex generalized associated Lame potential V(x)= a(a+1)m sn^2(y,m)+ b(b+1)m sn^2(y+K(m),m) + f(f+1)m sn^2(y+K(m)+iK'(m),m)+ g(g+1)m sn^2(y+iK'(m),m), where y = x-K(m)/2-iK'(m)/2, sn(y,m) is a Jacobi elliptic function with modulus parameter m, and there are four real parameters a,b,f,g. First, we derive two new duality relations which, when coupled with a previously obtained duality relation, permit us to relate the band edge eigenstates of the 24 potentials obtained by permutations of the four parameters a,b,f,g. Second, we pose and answer the question: how many independent potentials are there with a finite number "a" of band gaps when a,b,f,g are integers? For these potentials, we clarify the nature of the band edge eigenfunctions. We also obtain several analytic results when at least one of the four parameters is a half-integer. As a by-product, we also obtain new solutions of Heun's differential equation.
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"abstract": "We obtain several new results for the complex generalized associated Lame\npotential V(x)= a(a+1)m sn^2(y,m)+ b(b+1)m sn^2(y+K(m),m) + f(f+1)m\nsn^2(y+K(m)+iK\u0027(m),m)+ g(g+1)m sn^2(y+iK\u0027(m),m), where y = x-K(m)/2-iK\u0027(m)/2,\nsn(y,m) is a Jacobi elliptic function with modulus parameter m, and there are\nfour real parameters a,b,f,g. First, we derive two new duality relations which,\nwhen coupled with a previously obtained duality relation, permit us to relate\nthe band edge eigenstates of the 24 potentials obtained by permutations of the\nfour parameters a,b,f,g. Second, we pose and answer the question: how many\nindependent potentials are there with a finite number \"a\" of band gaps when\na,b,f,g are integers? For these potentials, we clarify the nature of the band\nedge eigenfunctions. We also obtain several analytic results when at least one\nof the four parameters is a half-integer. As a by-product, we also obtain new\nsolutions of Heun\u0027s differential equation.",
"arxiv_id": "quant-ph/0602105",
"authors": [
"Avinash Khare",
"Uday Sukhatme"
],
"categories": [
"quant-ph",
"math-ph",
"math.MP"
],
"doi": "10.1063/1.2204810",
"title": "Complex Periodic Potentials with a Finite Number of Band Gaps",
"url": "https://arxiv.org/abs/quant-ph/0602105"
},
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