Complex periodic potentials with a finite number of band gaps

We obtain several new results for the complex generalized associated Lame potential V(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≡x-K(m)/2-iK'(m)/2, sn(y,m) is the Jacobi elliptic function with modulus parameter m, and there are four real parameters a,b,f,g. First, we derive two new duality relations which, when coupled with a previously obtained duality relation, permit us to relate the band edge eigenstates of the 24 potentials obtained by permutations of the parameters a,b,f,g. Second, we pose and answer the question: how many independent potentials are there with a finite number 'a' of band gaps when a,b,f,g are integers and a≥b≥f≥g≥0? For these potentials, we clarify the nature of the band edge eigenfunctions. We also obtain several analytic results when at least one of the four parameters is a half-integer. As a by-product, we also obtain new solutions of Heun's differential equation