Analytically solvable PT-invariant periodic potentials

Associated Lame 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 --> ix + beta, where beta is any nonzero real number. These PT-invariant potentials are defined by V-PT(x) equivalent toV (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 VPT(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. (C) 2004 Elsevier B.V. All rights reserved.