Khare, Avinash ; Sukhatme, Uday (2006) Complex periodic potentials with a finite number of band gaps Journal of Mathematical Physics, 47 (6). 062103_1062103_22. ISSN 00222488

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Official URL: http://jmp.aip.org/resource/1/jmapaq/v47/i6/p06210...
Related URL: http://dx.doi.org/10.1063/1.2204810
Abstract
We obtain several new results for the complex generalized associated Lamé 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 = xK(m)/2iK'(m)/2, sn(y, m) is the 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 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 halfinteger. As a byproduct, we also obtain new solutions of Heun's differential equation.
Item Type:  Article 

Source:  Copyright of this article belongs to American Institute of Physics. 
Keywords:  Eigenvalues and Eigenfunctions; Differential Equations; Functional Analysis; Quantum Theory 
ID Code:  83288 
Deposited On:  20 Feb 2012 06:34 
Last Modified:  19 May 2016 00:12 
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