Completeness of coherent state subsystems for nilpotent Lie groups

TL;DR

This paper investigates the conditions under which subsystems of coherent states form complete bases in nilpotent Lie groups, linking the cyclicity of restricted representations to the completeness of these subsystems.

Contribution

It establishes a connection between the cyclicity of restricted representations and the completeness of coherent state subsystems for nilpotent Lie groups, providing necessary density conditions.

Findings

Necessary density conditions for completeness are derived.

Cyclicity of restricted representations influences subsystem completeness.

Results apply to coherent states on homogeneous G-spaces.

Abstract

Let $G$ be a nilpotent Lie group and let $\pi$ be a coherent state representation of $G$. The interplay between the cyclicity of the restriction $\pi|_{\Gamma}$ to a lattice $\Gamma \leq G$ and the completeness of subsystems of coherent states based on a homogeneous $G$-space is considered. In particular, it is shown that necessary density conditions for Perelomov's completeness problem can be obtained via density conditions for the cyclicity of $\pi|_{\Gamma}$.