Quasiperiodic media is a class of almost periodic media which is generated from periodic media through a ‘cut and project’ procedure. Quasiperiodic media displays some extraordinary optical, electronic and conductivity properties which call for the development of methods to analyse their microstructures and effective behaviour. In this paper, we develop the method of Bloch wave homogenisation for quasiperiodic media. Bloch waves are typically defined through a direct integral decomposition of periodic operators. A suitable direct integral decomposition is not available for almost periodic operators. To remedy this, we lift a quasiperiodic operator to a degenerate periodic operator in higher dimensions. Approximate Bloch waves are obtained for a regularised version of the degenerate operator. Homogenised coefficients for quasiperiodic media are obtained from the first Bloch eigenvalue of the regularised operator in the limit of regularisation parameter going to zero. A notion of quasiperiodic Bloch transform is defined and employed to obtain homogenisation limit for an equation with highly oscillating quasiperiodic coefficients.

中文翻译：

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准周期介质的布洛赫波均匀化

准周期性媒体是一类几乎周期性的媒体，它是通过“剪切和投影”程序从周期性媒体生成的。准周期介质表现出一些非凡的光学、电子和导电特性，需要开发分析其微观结构和有效行为的方法。在本文中，我们开发了准周期介质的布洛赫波均匀化方法。布洛赫波通常通过周期性算子的直接积分分解来定义。对于几乎周期性的运算符，没有合适的直接积分分解。为了解决这个问题，我们将准周期算子提升为更高维度的退化周期算子。获得了退化算子的正则化版本的近似布洛赫波。在正则化参数变为零的情况下，从正则化算子的第一个 Bloch 特征值获得准周期介质的均质化系数。定义并采用准周期布洛赫变换的概念来获得具有高度振荡准周期系数的方程的均匀化极限。