Square root of Dirac delta Function?

Question

I've puzzled by the appeareance and manipulation of a Square root of a Dirac Delta function.

The article is

https://www.researchgate.net/publication/243474570_Noncyclic_Geometric_Phase_Shift_for_Quantal_Revivals

and the relevant equation is not-numbered, but it is between eq (15) and (16) of the article.

So, what's the square root of a Dirac delta function and how do you even take derivatives of it?