Spectral Analysis of the local Commutator Operators
Abstract
Description
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.
16 pages, plain TeX
16 pages, plain TeX