Analytically solvable PT-invariant periodic potentials

Abstract

Associated Lame potentials V(x)=a(a+1)msn²(x,m)+b(b+1)mcn²(x,m)/dn²(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.