We consider a Bose gas in a one-dimensional periodic background formed by generalized delta function potentials with one and two impurities. We investigate the scattering off these impurities and their bound state levels. Besides expected features, we observe a kind of long-range correlation between the bound state levels of two impurities. Further, we define and calculate the vacuum energy of the impurity. It causes a force acting on the impurity relative to the background. We define the vacuum energy as a mode sum. In order to get a discrete spectrum we start from a finite lattice and use Chebychev polynomials to get a general expression. These allow also for quite easy investigation of impurities in finite lattices.

中文翻译：

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一维周期性背景中杂质的卡西米尔效应

我们考虑一维周期性背景下的Bose气体，该背景由具有一种和两种杂质的广义增量函数势形成。我们研究了这些杂质及其结合态水平的散射。除了预期的特征，我们还观察到了两种杂质的结合态能级之间的一种长程相关性。此外，我们定义并计算杂质的真空能。它导致相对于本底作用在杂质上的力。我们将真空能定义为众数。为了获得离散频谱，我们从有限晶格开始，并使用Chebychev多项式获得一般表达式。这些还允许非常容易地研究有限晶格中的杂质。