PT - invariant periodic potentials with a finite number of band gaps

Khare, Avinash ; Sukhatme, Uday (2005) PT - invariant periodic potentials with a finite number of band gaps Journal of Mathematical Physics, 46 (8). 082106_1-082106_18. ISSN 0022-2488

PDF - Author Version

Official URL:

Related URL:


We obtain the band edge eigenstates and the midband states for the complex, generalized associated Lamé potentials VPT(x) =-a(a+1) m sn2(y,m)-b(b+1) m sn2(y+K(m),m)-f(f+1)m sn2(y+K(m)+iK'(m),m)-g(g+1)m sn2(y+iK'(m),m), where y = ix+β, and there are four parameters a, b, f, g. By construction, this potential is PT-invariant 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 sn2(x,m)+b(b+1) m sn2(x+K(m),m) and their corresponding PT-invariant counterparts VPT(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 PT-invariant potentials VPT(x) are periodic problems with a finite number of band gaps. Further, using supersymmetry, we construct several additional, complex, PT-invariant, 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

Repository Staff Only: item control page