I'm going to ask a very vague question, and then give specifics for the version I particularly care about. I'm interested in answers at all levels of vagueness.


At the most vague version, I am in the following situation. I have some particular (smooth) function $A(t\_0,q\_0,t\_1,q\_1)$, where $t\_0,t\_1$ are real variables and $q\_0,q\_1$ vary over some manifold $M$, i.e. $A: \mathbb R \times M \times \mathbb R \times M \to \mathbb R$. I have another (smooth) function $Q(t\_0,q\_0,t\_1,q\_1;t) : \mathbb R \times M \times \mathbb R \times M \times \mathbb R \to M$.
 I am interested in (smooth) solutions $F(t\_0,q\_0,t\_1,q\_1)$ to the "composition" formula:
$$ A(t\_0,q\_0,t\_1,q\_1)\,F(t\_0,q\_0,t,Q(t\_0,q\_0,t\_1,q\_1;t))\,F(t,Q(t\_0,q\_0,t\_1,q\_1;t),t\_1,q\_1) = F(t\_0,q\_0,t\_1,q\_1)$$
I know one such (nonzero) solution explicitly. I am hoping that, presumably with slightly more data (some sort of "initial data"), I can conclude that my solution is uniquely determined. So my question is:



> 
> What extra data for $F$, and what restrictions on $A$, are needed for a functional equation of the above type to have a unique solution?
> 
> 
> 




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For those interested, I can give you more precise information. I have some "physics" encoded by a function $S(t\_0,q\_0,t\_1,q\_1)$, which I am thinking of as a "Hamilton-Jacobi function". This function defines $Q$ via:
$$ \left. \frac{\partial}{\partial q} \bigl[ S(t\_0,q\_0,t,q) + S(t,q,t\_1,q\_1) \bigr] \right]\_{q = Q(t\_0,q\_0,t\_1,q\_1;t)} = 0 $$
and satisfies:
$$ S(t\_0,q\_0,t,q) + S(t,q,t\_1,q\_1) \bigr|\_{q = Q(t\_0,q\_0,t\_1,q\_1;t)} = S(t\_0,q\_0,t\_1,q\_1)$$
The function $A$ is given by
$$ \left| \det \left[ \frac{\partial^2}{\partial q^2} \bigl[ S(t\_0,q\_0,t,q) + S(t,q,t\_1,q\_1) \bigr] \right]\_{q = Q(t\_0,q\_0,t\_1,q\_1;t)} \right|^{-1/2} $$
Then the function $F$ is:
$$ F(t\_0,q\_0,t\_1,q\_1) = \left| \det \frac{\partial^2}{\partial q\_0\partial q\_1} \bigl[ S(t\_0,q\_0,t\_1,q\_1) \bigr] \right|^{1/2}$$
which as defined makes sense only on $\mathbb R^n$. Actually, $A$ makes sense as a volume form, and $F$ as a half density in each variable, and then everything transforms correctly under changes of coordinates.


Anyway, I would really like to conclude that this $F$ is the only solution, but I don't know if I can.



What kind of uniqueness can I conclude for solutions to a simple functional equation?
Here's a vague answer then: there's [an old 1978 paper in Aequationes Mathematicae](https://doi.org/10.1007/BF01818567 "Baron, K., Sablik, M. On the uniqueness of continuous solutions of a functional equation of n-th order. Aequat. Math. 17, 295–304 (1978). https://zbmath.org/?q=an:0393.39002") mentioning the problem of uniqueness for the functional equation $\varphi (x)=h(x,\varphi [f\_1(x)],\dots, \varphi [f\_n(x)] )$ under some hypotheses. This seems to be of the same form as your equation (identifying $x=(q\_0,t\_0,q\_1,t\_1)$ and so on). Maybe what you're looking for is in there, or in later papers quoting that one.


Having said that, since your inspiration is Hamiltonian mechanics, I would imagine at least some variant of your problem to have been investigated already. Obviously you're looking at the stationary phase approximation to the quantum propagator, such as in [equation (33.32) of a chapter about the semiclassical propagator](https://chaosbook.org/chapters/VanVleck.pdf) in a [great physics textbook](https://chaosbook.org/). Now there's been a lot of mathematical work on the topic which may, at least implicitly, help you with your specific question, namely [this paper of Meinrenken](https://lib-extopc.kek.jp/preprints/PDF/1992/9201/9201019.pdf "Meinrenken, E. Semiclassical principal symbols and Gutzwiller's trace formula. Rep. Math. Phys. 31, No. 3, 279–295 (1992). doi:10.1016/0034-4877(92)90019-W. zbMATH review at https://zbmath.org/?q=an:0794.58046") in particular page 7 and beyond, and [that paper](https://web.ma.utexas.edu/mp_arc-bin/mpa?yn=97-599 "Combescure, M., Ralston, J. & Robert, D. A Proof of the Gutzwiller Semiclassical Trace Formula Using Coherent States Decomposition. Comm Math Phys 202, 463–480 (1999). doi:10.1007/s002200050591. https://zbmath.org/?q=an:0939.58031") of Combescure–Ralston–Robert.


