  Adiabaticity occurs when, during its evolution, a physical system remains in
the instantaneous eigenstate of the hamiltonian. Unfortunately, existing
results, such as the quantum adiabatic theorem based on a slow down evolution
(H(epsilon t), epsilon ? 0), are insufficient to describe an evolution driven
by the hamiltonian H(t) itself. Here we derive general criteria and exact
bounds, for the state and its phase, ensuring an adiabatic evolution for any
hamiltonian H(t). As a corollary we demonstrate that the commonly used
condition of a slow hamiltonian variation rate, compared to the spectral gap,
is indeed sufficient to ensure adiabaticity but only when the hamiltonian is
real and non oscillating (for instance containing exponential or polynomial but
no sinusoidal functions).
