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

We obtain the band edge eigenstates and the midband states for the complex, generalized associated Lame 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'x(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 Lame 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 Lame potential problem and Heun's differential equation