  This paper defines symplectic scale manifolds based on
Hofer-Wysocki-Zehnder's scale calculus. We introduce Hamiltonian vector fields
and flows on these by narrowing down sc-smoothness to what we denote by strong
sc-smoothness, a concept which effectively formalizes the desired smoothness
properties for Hamiltonian functions. We show the concept to be invariant under
sc-smooth symplectomorphisms, whence it is compatible with Hofer's scale
manifolds. We develop and verify the theory at the hand of the free
Schr\"odinger equation.
