Unitary equivalence of lowest dimensional reproducing formulae of type ${\mathcal{E}}_2\subset Sp(2,\mathbb{R})$

Abstract

All two-dimensional reproducing formulae, i.e. of $L^2({\mathbb{R}}^2)$, resulting out of restrictions of the projective metaplectic representation to connected Lie subgroups of $Sp(2,\mathbb{R})$ and of type ${\mathcal{E}}_2$, were listed and classified up to conjugation within $Sp(2,\mathbb{R})$ in [2], [3]. A full classification, up to conjugation within ${\mathbb{R}}^2⋊Sp(1,\mathbb{R})$, of one-dimensional reproducing formulae, i.e. of $L^2(\mathbb{R})$, resulting out of restrictions of the extended projective metaplectic representation to connected Lie subgroups of ${\mathbb{R}}^2⋊Sp(1,\mathbb{R})$ was obtained in [13], [14]. In dimension one, there are no reproducing formulae with one-dimensional parametrizations, yet in dimension two, there are reproducing formulae with two-dimensional parametrizations. Two-dimensional reproducing subgroups of $Sp(2,\mathbb{R})$ of type ${\mathcal{E}}_2$ are a novelty. They exhibit completely new phase space phenomena. We show, that they are all unitarily equivalent via natural choices of coordinate systems, and we derive the consequences of this equivalence.

21 pages