NC plane waves, Casimir effect and flux tube potential with L\"uscher terms

We analyze plane waves in a model of quantum mechanics in a three dimensional noncommutative (NC) space $R^3_{\lambda}$. Signature features of NC models are impossibility of probing distances smaller than a certain length scale {\lambda} and a presence of natural energetic cut-off at energy scale of order $1/{\lambda}^2$ (in convenient units). We analyze consequences of such restrictions on a 1 dimensional Casimir effect. The result shows resemblance to flux tube potential for quark-antiquark pairs and to effective bosonic string theories with L\"uscher terms. Such behavior might effect the radius of possible compact (fuzzy) dimensions.