Abstract: We study Schr\"odinger operators with a $\delta$ interaction supported by spheres. First we discuss a tunneling decay of $\delta$-confined states and show that if the initial state is described by a constant function within the sphere, the decay law has a peculiar stairlike shape. In the second part we generalize results of R. Hempel et al. to the case of infinitely many $\delta$-interactions on the concentric spheres and show that the positive spectrum consists of alternating bands of absolutely continuous and dense pure point character. The seminar summarizes a diploma thesis.