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

Applying certain known theorems about one-dimensional periodic potentials, we show that the energy spectrum of the associated Lam\'{e} potentials $$a(a+1)m~{\rm sn}^2(x,m)+b(b+1)m~{\rm cn}^2(x,m)/{\rm dn}^2(x,m)$$ 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 ($a \ne b$), 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.