(Université de Varsovie)
Abstract: 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 one- dimensional Schrödinger operators with inverse square potential and 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 and induces a simple but nontrivial ow in the parameter space. One can say (with some exaggeration) that they can be viewed as « toy models of the renormalization group ».