Relativistic Lame functions: the special case g = 2

We study a class of eigenfunctions of an analytic difference operator gener alizing the special Lame operator -d(2)/dx(2) + 2(sic) (x), paying particul ar attention to quantum-mechanical aspects. We show that in a suitable scal ing limit the pertinent eigenfunctions lend to the eigenfunctions of the op erator -d(2)/dx(2) + 2c(sic)(x) in a finite volume. We establish various or thogonality and non-orthogonality results by direct calculations, generaliz e the `one-gap picture' associated with the above Lame operator, and obtain duality properties for the hyperbolic, trigonometric and rational speciali zations.