  Trace formulae provide one of the most elegant descriptions of the
classical-quantum correspondence. One side of a formula is given by a trace of
a quantum object, typically derived from a quantum Hamiltonian, and the other
side is described in terms of closed orbits of the corresponding classical
Hamiltonian. In algebraic situations, such as the original Selberg trace
formula, the identities are exact, while in general they hold only in
semi-classical or high-energy limits. We refer to a recent survey \cite{Ur} for
an introduction and references.
  In this paper we present an intermediate trace formula in which the original
trace is expressed in terms of traces of quantum monodromy operators directly
related to the classical dynamics. The usual trace formulae follow and in
addition this approach allows handling effective Hamiltonians.
