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.
Categories: quant-ph cond-mat hep-th math-ph math.MP