First, I will describe a certain natural holomorphic family of closed operators with interesting spectral properties. These operators can be fully analyzed using just trigonometric functions.

Then, I will discuss 1-dimensional Schrödinger operators with a x^-2 potential with general boundary conditions, which I studied recently with S. Richard. Even though their description involves Bessel and Gamma functions, they turn out to be equivalent to the previous family.

Some operators that I will describe are homogeneous — they get multiplied by a constant after a change of the scale. In general, their homogeneity is weakly broken — scaling induces a simple but nontrivial flow in the parameter space. One can say (with some exaggeration) that they can be viewed as "toy models of the renormalization group".