Bi-coherent states as generalized eigenstates of the position and the momentum operators

TL;DR

This paper introduces bi-coherent states as generalized eigenstates of position and derivative operators, connecting biorthogonal families of monomials and delta derivatives, extending the concept of coherent states.

Contribution

It presents a novel framework for bi-coherent states based on biorthogonal families linked to position and derivative operators, generalizing traditional coherent state theory.

Findings

Biorthogonal families connect position and derivative operators.

Bi-coherent states can be constructed as series or via displacement operators.

The approach generalizes known coherent state results.

Abstract

In this paper we show that the position and the derivative operators, $\hat q$ and $\hat D$, can be treated as ladder operators connecting the various vectors of two biorthonormal families, $\mathcal{F}_\varphi$ and $\mathcal{F}_\psi$. In particular, the vectors in $\mathcal{F}_\varphi$ are essentially monomials in $x$, $x^k$, while those in $\mathcal{F}_\psi$ are weak derivatives of the Dirac delta distribution, $\delta^{(m)}(x)$, times some normalization factor. We also show how bi-coherent states can be constructed for these $\hat q$ and $\hat D$, both as convergent series of elements of $\mathcal{F}_\varphi$ and $\mathcal{F}_\psi$, or using two different displacement-like operators acting on the two vacua of the framework. Our approach generalizes well known results for ordinary coherent states.