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

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, $\mathcal{F}_φ$ and $\mathcal{F}_ψ$. In particular, the vectors in $\mathcal{F}_φ$ are essentially monomials in $x$, $x^k$, while those in $\mathcal{F}_ψ$ are weak derivatives of the Dirac delta distribution, $δ^{(m)}(x)$, 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 $\mathcal{F}_φ$ and $\mathcal{F}_ψ$, 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.