The One-Dimensional Alternative Pseudodifferential Analysis

Abstract

In this chapter, we introduce and study alternative pseudodifferential analysis, i.e., pseudodifferential analysis in connection with anaplectic analysis on the line. One of its most characteristic features is that it splits into an ascending and a quite similar descending parts: we shall concentrate on the first one. Under any operator from the ascending calculus, an eigenstate of the (standard or not) harmonic oscillator L _z transforms into the sum of a series of eigenstates of L _z with higher energy level. Section 3.1 introduces a formal definition of the ascending calculus and proves its covariance properties, a true but not “manifest” one when Heisenberg representation is concerned: in a remark at the end of the same section, we shall explain the geometric ideas that led to this definition.