Jan Derezinsky

(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 ».

Based on

• J.D. Laurent Bruneau and Vladimir Georgescu: Homogeneous Schrödinger operators on half-line, Annales Henri Poincaré 12 (2011), 547-590 ;

• J.D., Serge Richard: On Schrödinger operators with inverse square potentials on the half-line, Annales Henri Poincaré 18 (2017) 869-928;

• J.D.: Homogeneous rank one perturbations, to appear in Annales Henri Poincaré