无公度超晶格和准晶超空间中的超哈密顿量
Super Hamiltonian in superspace for incommensurate superlattices and quasicrystals
摘要：
空间中的无限准周期性排列，如准晶，通常被描述为高维周期性晶格在物理维度上的投影。一个参考高维空间的概念，称为超空间，已被证明是有用的准周期系统。虽然一些准周期介质中的量子力学系统已经被证明可以存在准周期态，但到目前为止，在超空间中还缺乏任何形式的一般哈密顿形式。在这里，我们展示了如何将一般的量子力学哈密顿量扩展到更高的维度，使得原始哈密顿量的本征态作为哈密顿量在超空间中的投影而获得，我们称之为超哈密顿量。我们将超哈密顿形式应用于一个简单而又现实的一维量子粒子，在没有紧束缚近似的情况下，在准周期势中得到了对应于连续谱的连续标记本征态.所有对应于连续统的态都是准周期的。我们还得到了封闭形式的连续态的格林函数，并由此得到了态密度和局域态密度，以及缺陷和杂质的散射态。这个一维格林函数的封闭形式对于任何一维单粒子量子系统中的任何连续态都是同样有效的，该系统允许连续谱。与我们使用的基组，这是周期性的超空间，因此准周期性的物理空间，我们发现，安德森局域化的状态也准周期性的，如果分配的解决方案是承认，但规避这一困难，通过推广超空间方法开放的边界条件（x'd）。
英文摘要：
Infinite quasiperiodic arrangements in space, such as quasicrystals, are typically described as projections of higher-dimensional periodic lattices onto the physical dimension. The concept of a reference higher-dimensional space, called a superspace, has proved useful in relation to quasiperiodic systems. Although some quantum-mechanical systems in quasiperiodic media have been shown to admit quasiperiodic states, any sort of general Hamiltonian formalism in superspace is lacking to this date. Here, we show how to extend generic quantum-mechanical Hamiltonians to higher dimensions in such a way that eigenstates of the original Hamiltonian are obtained as projections of the Hamiltonian in superspace, which we call the super Hamiltonian. We apply the super Hamiltonian formalism to a simple, yet realistic one-dimensional quantum particle in a quasiperiodic potential without the tight-binding approximation, and obtain continuously labelled eigenstates of the system corresponding to a continuous spectrum. All states corresponding to the continuum are quasiperiodic. We also obtain the Green's functions for continuum states in closed form and, from them, the density of states and local density of states, and scattering states off defects and impurities. The closed form of this one-dimensional Green's function is equally valid for any continuum state in any one-dimensional single-particle quantum system admitting continuous spectrum. With the basis set we use, which is periodic in superspace, and therefore quasiperiodic in physical space, we find that Anderson-localised states are also quasiperiodic if distributional solutions are admitted, but circumvent this difficulty by generalising the superspace method to open boundary conditions (cont'd).
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