Alternative Pseudodifferential Analysis pp 75-91 | Cite as

# From Anaplectic Analysis to Usual Analysis

## Abstract

It is possible to consider anaplectic analysis on the real line as a special case of a oneparameter family of analyses. The parameter \(\nu\) is a complex number mod 2, subject to the restriction that it should not be an integer: anaplectic analysis, as considered until now, corresponds to the case when \(\nu = - \frac{1}{2}\). There is a natural \(\nu\)-anaplectic representation of some cover of *SL*(2,ℝ) in some space \(\mathfrak{A}_\nu\), compatible in the usual way with the Heisenberg representation; the \(\nu\)-anaplectic representation is pseudounitarizable in the case when \(\nu\) is real. Depending on \(\nu\), the even or odd part of the \(\nu\)-anaplectic representation coincides in this case with a representation taken from the unitary dual of the universal cover of *SL*(2,ℝ), as completely described by Pukanszky [24]. In \(\nu\)-anaplectic analysis, the spectrum of the harmonic oscillator is the arithmetic sequence \(\nu + \frac{1}{2} + \mathbb{Z}\). Much of the theory subsists in the case when \(\nu \equiv 0 \) mod 2, which leads to a nontrivial enlargement of usual analysis. However, as we shall make clear, while the ascending pseudodifferential calculus extends to the case of \(\nu\)-anaplectic analysis when \(\nu \in \mathbb{C}\backslash\mathbb{Z}\), it is impossible to extend it to the usual analysis environment.

## Keywords

Harmonic Oscillator Universal Cover Principal Series Usual Analysis Hermite Function## Preview

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