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

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 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 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 PT-invariant potentials V^PT(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.