PT-Invariant Periodic Potentials with a Finite Number of Band Gaps

We obtain the band edge eigenstates and the mid-band states for the complex, PT-invariant 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. This work is a substantial generalization of previous work with the associated Lamé potentials V (x) = a(a+1)msn2(x,m) + b(b+1)msn2(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, new, 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.