  The aim of this paper is to develop a theory of microdifferential operators
for arithmetic $\mathscr{D}$-modules. We first define the sheaves of
microdifferential operators of arbitrary levels on arbitrary smooth formal
schemes. A difficulty lies in the fact that there are no homomorphisms between
sheaves of microdifferential operators of different levels. To remedy this, we
define the intermediate differential operators, and using these, we define the
sheaf of microdifferential operators for $\mathscr{D}^\dag$. We conjecture that
the characteristic variety of a $\mathscr{D}^\dag$-module is computed as the
support of the microlocalization of a $\mathscr{D}^\dag$-module, and prove it
in the curve case.
