A QES band-structure problem in one dimension

Abstract

I show that the potential V(x,m) = [ b²/4 - m(1-m)a(a+1) ] sn²(x,m)/ dn²(x,m) - b ( a + ½ ) cn(x,m)/dn² (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 bound state eigenvalues and eigenfunctions of the potential V(x) = [ β²/4 - a(a+1) ] sech² + β (a + ½ ) sechxtanhx in case a is any positive integer or half-integer.