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, q^ and D^ , can be treated as ladder operators connecting various vectors of two biorthonormal families, Fφ and Fψ. In particular, the vectors in Fφ are essentially monomials in x, xk, while those in Fψ 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 q^ and D^ , both as convergent series of elements of Fφ and Fψ, 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