伴随拉梅势的中带态和零带隙态的一些精确结果
Some Exact Results for Mid-Band and Zero Band-Gap States of Associated Lame Potentials
摘要：
应用一维周期势的某些已知定理，证明了当$b$和$a$都取整数值时，相应的Lamé势$$a(a+1)m~{\rm sn}^2(x,m)+b(b+1)m~{\rm cn}^2(x,m)/{\rm dn}^2(x,m)$$的能谱由有限个束缚带和一个连续带组成.进一步，如果$a$和$b$是不等的整数，我们证明了一定存在一些零带隙态，即具有相同节点数的双简并态.更一般地说，在$a$和$b$不是整数，但$a + b$或$a - b$是整数（$a \ne b$）的情况下，我们再次表明，由于具有相同节点数的状态的简并，几个带隙消失。最后，当$a$或$b$为整数，另一个取半整数值时，我们得到了几个中带态的精确解析解。
英文摘要：
Applying certain known theorems about one-dimensional periodic potentials, we show that the energy spectrum of the associated Lamé 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.
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