I'm looking for some help in understanding the PSS isomorphism map in the context of Hamiltonian Floer cohomology and Morse cohomology with universal Novikov coefficients $\Lambda_{\omega}$ (à la Seidel for example, i.e. I'm keeping track of the energy of the $J$-holomorphic strips defining the differential): Let $H^{\alpha}$ a non-degenerate Hamiltonian on $(M,\omega)$, let $f^{\beta}$ be a Morse function on $M$ whose gradient w.r.t. to the metric induced by the symplectic form and compatible a.c.s. is Morse-Smale. Let's say that $(M,\omega)$ is monotone on spheres, in order to have a well-defined Hamiltonian Floer cohomology without the need of advanced virtual techniques.
The map $\Phi^{\alpha}_{\beta}: CF^*(H^{\alpha},\Lambda_{\omega}) \to CM^*(f^{\beta},\Lambda_{\omega})$ which induces the PSS isomorphism is defined as a count of index $0$ "spiked disks" $(u,\gamma)$, i.e. finite-energy $J$-holomorphic planes $u$ with a marked point at the origin together with a gradient flow $\gamma: (-\infty,0]\to M$ such that $\gamma(0)=u(0)$ and $\lim_{s\to -\infty} \gamma = \tilde{x}$, critical point of $f^{\beta}$. How does it work with universal Novikov coefficients? My guess is that we need to keep track of the symplectic area of the disks $u$ and possibly the length of the trajectories $\gamma$, in order to ensure compactness of the the relevant moduli spaces (of $J$-hol. disks I presume)
Do we have a PSS isomorphism even in the case of other kind of coefficients? I'm thinking about the case of $\Lambda_{\omega}^0$, i.e. the Novikov ring of "generalised Laurent series$ but with positive exponents?
What kind of naturality should I expect to see in the PSS isomorphism? On McDuff-Salamon they claim that if I make the PSS land into quantum cohomology then the PSS isomorphism intertwines the continuation maps in the domain. On [Dur] (Theorem 2), the author proves (in their setting) that the PSS isomorphism intertwines continuation maps in the domain and codomain (i.e. morse continuation maps as defined by Schwarz in his Morse theory book). To me it looks like that's a standard argument and doesn't really relies on their specific setting. I'm surprised I haven't seen it written on other references though.
I apologise in advance if these are ill-posed question, I'm browsing the literature and trying to piece together all these informations.
[Dur] J. Duretic - Piunikhin-Salamon-Schwarz isomorphisms and spectral invariants for conormal bundle - Publ. Inst. Math. (Beograd) (N.S.) 102(116) (2017), 17--47
