# $kq$-representation for pseudo-bosons, and completeness of bi-coherent states

## TL;DR

This paper adapts the Zak $kq$-representation to pseudo-bosons and demonstrates the completeness of bi-coherent states, including Riesz bi-coherent states, using this framework.

## Contribution

It introduces a method to apply the Zak $kq$-representation to pseudo-bosons and proves the completeness of associated bi-coherent states.

## Key findings

- Established conditions for the $kq$-representation with pseudo-bosons.
- Proved completeness of discrete bi-coherent states.
- Analyzed Riesz bi-coherent states in detail.

## Abstract

We show how the Zak $kq$-representation can be adapted to deal with pseudo-bosons, and under which conditions. Then we use this representation to prove completeness of a discrete set of bi-coherent states constructed by means of pseudo-bosonic operators. The case of Riesz bi-coherent states is analyzed in detail.