Let $(M,\omega)$ be a symplectic manifold and let $H$ a Hamiltonian function. If $M$ is not closed we consider $H$ to be linear at infinity to ensure that $HF^*(H)$ is well-defined (I'm particularly interested in this non-closed case).
The PSS map (not necessarily an isomorphism for $M$ non-closed) $$\Phi:QH^*(M)\to HF^*(H)$$ is known to be a ring isomorphism mapping the quantum cup product to the pair of pants product.
What is known about the behavior of $\Phi$ when the quantum cup product is the ordinary cup product? Explicitly, is a general formula known for $$ \Phi(a\smile b) $$ for $a,b\in H^*(M)$?
