With the introduction of the vector $ \mathbf {A} $ and scalar $ \varphi $ of the potentials of the electromagnetic field, an ambiguity arises that does not create any problems of a fundamental nature, but requires resolution in order to carry out calculations in specific problems. Namely, the transformation
$$ \mathbf {A} \rightarrow \mathbf {A} +\nabla \psi, $$
$$ \varphi \rightarrow \varphi - \frac {\partial \psi }{\partial t}, $$
where $ \psi =\psi (\vec {r},t) $ is an arbitrary scalar function of coordinates $ \vec {r} $ and time $ t $, do not change the form of Maxwell's equations, and hence are admissible from a physical point of view. It is necessary to dwell on some choice of this function, and it can be made for reasons of mathematical convenience.
But in practice, the function is not fixed $ \psi $ (with previously introduced potentials), but the imposition of some additional condition on the potentials themselves.
So my question, is it possible to set the function view for a specific gauge? Say for Coulomb gauge $$\mathrm{div}\mathbf{A} = 0 $$ or for Lorenz gauge $$\mathrm{div}\,\mathbf{A} + {1 \over c^2}{\partial \mathbf{\varphi} \over \partial t} = 0~?$$
