The energy spectrum of complex periodic potentials of the Kronig-Penney type

We consider a complex periodic PT-symmetric potential of the Kronig-Penney type, in order to elucidate the peculiar properties found by Bender et al. for potentials of the form V = i(sin x)2N+1, and in particular the absence of anti-periodic solutions. In this model we show explicitly why these solutions disappear as soon as V (x) 6= V (x), and spell out the consequences for the form of the dispersion relation. In a recent paper Bender et al. [1] showed that periodic potentials which were complex but obeyed PT symmetry possessed real band spectra, with, however, one striking difference from the case of real periodic potentials, namely that there were no antiperiodic solutions, i.e. Bloch waves with lattice wave vector k = (2n + 1)π/a. This result was obtained from detailed numerical studies of potentials of the form V (x) = i sin(x), and it was found necessary to work to extremely high accuracy to detect the absence of such solutions. In the present note we supplement this work by an analytical solution to a complex, PT-symmetric version of the Kronig-Penney model, which illustrates the phenomenon very clearly. It therefore seems a generic property of nonHermitian, but PT symmetric potentials, although an analytic proof is still not available. In the standard Kronig-Penney model [2] the potential consists of a periodic string of delta functions of the form V (x) = α ∑