Analytically Solvable PT-Invariant Periodic Potentials

Associated Lam\'e potentials $V(x)=a(a+1)m\sn^2(x,m)+b(b+1)m{\cn^2 (x,m)}/{\dn^2(x,m)}$ are used to construct complex, PT-invariant, periodic potentials using the anti-isospectral transformation $x \to ix+\beta$, where $\beta$ is any nonzero real number. These PT-invariant potentials are defined by $V^{PT}(x) \equiv -V(ix+\beta)$, and have a different real period from $V(x)$. They are analytically solvable potentials with a finite number of band gaps, when $a$ and $b$ are integers. Explicit expressions for the band edges of some of these potentials are given. For the special case of the complex potential $V^{PT}(x)=-2m\sn^2(ix+\beta,m)$, we also analytically obtain the dispersion relation. Additional new, solvable, complex, PT-invariant, periodic potentials are obtained by applying the techniques of supersymmetric quantum mechanics.