Abstract
I show that the potential V(x,m) = \big [\frac{b^2}{4}-m(1-m)a(a+1) \big ]\frac{\sn^2 (x,m)}{\dn^2 (x,m)} -b(a+{1/2}) \frac{\cn (x,m)}{\dn^2 (x,m)} constitutes a 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. 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) = \big [\frac{\beta^2}{4}-a(a+1) \big ] \sech^2 x +\beta(a+{1/2})\sech x\tanh x in case a is any positive integer or half-integer.