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

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 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.