TITLE: Are there n-th roots of differential operators?
QUESTION [9 upvotes]: In analogy to a Dirac operator, it seems to me that formally, the equation
$$\frac{\partial^n}{\partial x^n}f(x,y)=D_yf(x,y)$$
is solved by 
$$f(x,y)=\exp{(x \sqrt[n]{D_y})}\ g(y).$$
Is there a theory surronding the $\sqrt[n]{D_y}$-idea?

REPLY [2 votes]: Since you mentioned the Dirac operator, which works not by analysis but by extending the scalars in a noncommutative way; consider
$$\left( \begin{array}{ccc} 0 & 0 & D_y  \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{array} \right)^3 $$
and generalize.