有限个带隙的复周期势
Complex Periodic Potentials with a Finite Number of Band Gaps
摘要：
本文对复广义缔合拉梅势V（x）= a（a+1）msn^2（y，m）+ b（b+1）msn^2（y+K（m），m）+ f（f+1）msn^2（y+K（m）+iK '（m），m）+ g（g+1）msn^2（y+ iK'（m），m）得到了几个新的结果，其中y = x-K（m）/2-iK '（m）/2，sn（y，m）是一个模参数为m的Jacobi椭圆函数，有四个实参数a，b，f，g首先，我们推导出两个新的对偶关系，当与先前获得的对偶关系相结合时，允许我们将通过四个参数a，b，f，g的排列获得的24个势的带边本征态联系起来。其次，我们提出并回答了这样一个问题：当a，b，f，g为整数时，有多少个独立的势函数具有有限数量的带隙“a”？对于这些潜在的，我们澄清的性质的带边本征函数。当四个参数中至少有一个是半整数时，我们也得到了几个解析结果。作为副产品，我们还获得了Heun微分方程的新解。
英文摘要：
We obtain several new results for the complex generalized associated Lame potential V(x)= a(a+1)m sn^2(y,m)+ b(b+1)m sn^2(y+K(m),m) + f(f+1)m sn^2(y+K(m)+iK'(m),m)+ g(g+1)m sn^2(y+iK'(m),m), where y = x-K(m)/2-iK'(m)/2, sn(y,m) is a Jacobi elliptic function with modulus parameter m, and there are four real parameters a,b,f,g. First, we derive two new duality relations which, when coupled with a previously obtained duality relation, permit us to relate the band edge eigenstates of the 24 potentials obtained by permutations of the four parameters a,b,f,g. Second, we pose and answer the question: how many independent potentials are there with a finite number "a" of band gaps when a,b,f,g are integers? For these potentials, we clarify the nature of the band edge eigenfunctions. We also obtain several analytic results when at least one of the four parameters is a half-integer. As a by-product, we also obtain new solutions of Heun's differential equation.
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