Some exact results for mid-band and zero band-gap states of associated Lamé potentials

Abstract

Applying certain known theorems about one-dimensional periodic potentials, we show that the energy spectrum of the associated Lamé potentials, a(a+1)m sn²(x,m)+b(b+1)m cn²(x,m)/dn²(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 ≠ 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.