Casimir effect for impurity in periodic background in one dimension

Abstract

Abstract 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.