Khare, Avinash ; Sukhatme, Uday (2005) PT  invariant periodic potentials with a finite number of band gaps Journal of Mathematical Physics, 46 (8). 082106_1082106_18. ISSN 00222488

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Official URL: http://jmp.aip.org/resource/1/jmapaq/v46/i8/p08210...
Related URL: http://dx.doi.org/10.1063/1.2000207
Abstract
We obtain the band edge eigenstates and the midband states for the complex, generalized associated Lamé potentials V^{PT}(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 = ix+β, and there are four parameters a, b, f, g. By construction, this potential is PTinvariant since it is unchanged by the combined parity (P) and time reversal (T) transformations. This work is a substantial generalization of previous work with the associated Lamé potentials V(x) =a(a+1)m sn^{2}(x,m)+b(b+1) m sn^{2}(x+K(m),m) and their corresponding PTinvariant counterparts V^{PT}(x) = V(ix+β), both of which involving just two parameters a,b. We show that for many integer values of a,b,f,g, the PTinvariant potentials V^{PT}(x) are periodic problems with a finite number of band gaps. Further, using supersymmetry, we construct several additional, complex, PTinvariant, periodic potentials with a finite number of band gaps. We also point out the intimate connection between the above generalized associated Lamé potential problem and Heun's differential equation.
Item Type:  Article 

Source:  Copyright of this article belongs to American Institute of Physics. 
Keywords:  Differential Equations; Supersymmetry; Eigenvalues and Eigenfunctions; Quantum Theory 
ID Code:  83285 
Deposited On:  20 Feb 2012 06:34 
Last Modified:  19 May 2016 00:11 
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