Spectral Analysis of the local Commutator Operators

Abstract

The spectral analysis of the (local) conductor operator H = log(|q|) + log(|p|) was shown in a previous paper to be given by the Explicit Formula. I give here the spectral analysis of the commutator operator K = i[log(|p|),log(|q|)] (which shares with H the property of complete dilation invariance). Its spectral function is found to be the derivative of the one of H. It is then deduced that K anticommutes with the Inversion and is a bounded operator. Higher commutators are related in the same manner to the higher derivatives of the Gamma function, which thus all acquire an operator theoretic and spectral interpretation.