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

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We obtain the band edge eigenstates and the mid-band states for the complex, PT-invariant generalized associated Lamé potentials VPT(x)=-a(a+1)m ²(y,m)-b(b+1)m ² (y+K(m),m) -f(f+1)m ² (y+K(m)+iK'(m),m)-g(g+1)m ² (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)m²(x,m)+b(b+1)m² (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, usingsupersymmetry, we construct several additional, new, 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.