A QES band-structure problem in one dimension

I show that the potential V(x,m)=[ b(2)/4-m(1-m)a(a+1)]sn(2)(x,m)/dn(2)(x,m)-b(a+1/2)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 e igenvalues and eigenfunctions can be obtained analytically. In the limit of in going to 0 or 1, I recover the well known results for the QES double si ne-Gordon or double sinh-Gordon equations, respectively. As a by product, I also obtain the bound state eigenvalues and eigenfunctions of the potentia l V(x)=[beta (2)/4-a(a+1)]sech(2)x+beta (a+1/2)sechxtanhx in case a is any positive integer or half-integer. (C) 2001 Published by El sevier Science B.V.