On the completeness of a system of coherent states

TL;DR

This paper proves the completeness of certain subsystems of coherent states, investigates their linear dependence, and characterizes the unique linear relation in the case of a regular lattice with cell area π, linking it to theta function identities.

Contribution

It establishes the completeness of specific coherent state subsystems and characterizes their linear dependence, especially for lattice-based states, connecting to theta function identities.

Findings

Completeness is proved for some subsystems of coherent states.

In the lattice case, only one linear relation exists among the states.

The linear relation is equivalent to identities involving theta functions.

Abstract

Completeness is proved for some subsystems of a system of coherent states. The linear dependence of states is investigated for the von Neumann type subsystems. A detailed study is made of the case when a regular lattice on the complex $\alpha$ plane with cell area S=$\pi$ corresponds to the states of the system. It is shown that in this case there exists only one linear relationship between the coherent states. This relationship is equivalent to an infinite set of identities. The symplest of these can also be obtained by means of the transformation formulas for $\theta$ functions.