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arXiv math-ph/0505027math-phhep-thmath.MPquant-ph
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有限个带隙的PT不变周期势

PT-Invariant Periodic Potentials with a Finite Number of Band Gaps

Avinash Khare, Uday Sukhatme

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30 pages, 0 figures
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摘要：

我们得到了复杂的PT不变广义关联Lamé势$V^{PT}(x)=-a(a+1)m \sn^2(y,m)-b(b+1)m {\sn^2 (y+K(m),m)} -f(f+1)m {\sn^2 (y+K(m)+iK'(m),m)}-g(g+1)m {\sn^2 (y+iK'(m),m)}$的带边本征态和中带态，其中$y \equiv ix+β$，有四个参数$a,b,f,g$ .这项工作是以前的工作与相关的Lamé势$V(x)=a(a+1)m\sn^2(x,m)+b(b+1)m{\sn^2 (x+K(m),m)}$和它们相应的PT不变的副本$V^{PT}(x)=-V(ix+β)$，这两个都只涉及两个参数$a,b$的实质性推广。我们表明，对于许多整数值的$a,b,f,g$，PT不变的潜力$V^{PT}(x)$是周期性的问题，有限数量的带隙。此外，利用超对称性，我们构建了几个额外的，新的，复杂的，PT不变的，周期性的潜力与有限数量的带隙。同时指出了广义伴随Lamé势问题与Heun微分方程之间的密切联系。

英文摘要：

We obtain the band edge eigenstates and the mid-band states for the complex, PT-invariant generalized associated Lamé potentials $V^{PT}(x)=-a(a+1)m \sn^2(y,m)-b(b+1)m {\sn^2 (y+K(m),m)} -f(f+1)m {\sn^2 (y+K(m)+iK'(m),m)}-g(g+1)m {\sn^2 (y+iK'(m),m)}$, where $y \equiv ix+β$, and there are four parameters $a,b,f,g$. This work is a substantial generalization of previous work with the associated Lamé potentials $V(x)=a(a+1)m\sn^2(x,m)+b(b+1)m{\sn^2 (x+K(m),m)}$ and their corresponding PT-invariant counterparts $V^{PT}(x)=-V(ix+β)$, both of which involving just two parameters $a,b$. We show that for many integer values of $a,b,f,g$, the PT-invariant potentials $V^{PT}(x)$ are periodic problems with a finite number of band gaps. Further, usingsupersymmetry, we construct several additional, new, complex, PT-invariant, periodic potentials with a finite number of band gaps. We also point out the intimate connection between the above generalized associated Lamé potential problem and Heun's differential equation.

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