A Variational Approach To Complex Periodic Potentials With Real Band Spectra

In this paper we study a class of complex \emph{PT}--symmetric periodic potentials possessing real band structures. In particular we shall investigate the potentials $V(x)=i\sin^{2N+1}(x)$ ($N=0,1,2,\ldots$) which are known to have infinitely many gaps. We note for such potentials, that at the band edges there are periodic wave functions with no anti-periodic ones. We will apply a recently developed variational ansatz wherein a finite (variational) basis is constructed with respect to a variational parameter $\lambda$, according to the schema $\psi_{n}=\partial^{n}_{\lambda}\psi_{0}\left( x,\lambda\right) $. Comparisons are then made to both numerical analysis as well as higher-order WKB techniques.