Some Exact Results for Mid-Band and Zero Band-Gap States of Associated Lamé Potentials

Applying certain known theorems about one-dimensional periodic potentials, we show that the energy spectrum of the associated Lamé potentials a(a+ 1)m sn(x,m) + b(b+ 1)m cn(x,m)/dn(x,m) consists of a finite number of bound bands followed by a continuum band when both a and b take integer values. Further, if a and b are unequal integers, we show that there must exist some zero band-gap states, i.e. doubly degenerate states with the same number of nodes. More generally, in case a and b are not integers, but either a + b or a − b is an integer (a 6= b), we again show that several of the band-gaps vanish due to degeneracy of states with the same number of nodes. Finally, when either a or b is an integer and the other takes a half-integral value, we obtain exact analytic solutions for several mid-band states. [email protected] [email protected]