An Unusual QES Band-Structure Problem in One Dimension

I show that the potential V (x,m) = [b 4 −m(1−m)a(a+ 1) ] sn(x,m) dn(x,m) − b(a+ 1 2 ) cn(x,m) dn(x,m) constitutes an unusual QES band-structure problem in one dimension. In particular, I show that for any positive integral or half-integral a, 2a+ 1 band edge eigenvalues and eigenfunctions can be obtained analytically. However, the associated orthogonal polynomials satisfy fourterm (rather than three-term ) recursion relation. In the limit of m going to 0 or 1, I recover the well known results for the QES double sine-Gordon or double sinh-Gordon equations respectively. As a by product, I also obtain the boundstate eigenvalues and eigenfunctions of the potential V (x) = [β 4 − a(a+ 1) ] sechx+ β(a+ 1 2 )sechx tanhx in case a is any positive integer or half-integer.