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, ${\mathcal{F}}_{\phi }$ and ${\mathcal{F}}_{\psi }$. In particular, the vectors in ${\mathcal{F}}_{\phi }$ 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 $\widehat{q}$ and $\widehat{D}$, both as convergent series of elements of ${\mathcal{F}}_{\phi }$ 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.