Analytically Solvable PT-Invariant Periodic Potentials

Associated Lamé potentials V (x) = a(a + 1)msn2(x,m) + b(b + 1)mcn2(x,m)/dn(x,m) are used to construct complex, PT-invariant, periodic potentials using the anti-isospectral transformation x→ ix+ β, where β is any nonzero real number. These PT-invariant potentials are defined by V PT (x) ≡ −V (ix + β), 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) = −2msn2(ix+ β,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.