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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?


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.

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Incidentally, checking the slightly more explicit case is a nice exercise, or see my papers on the arXiv :) In any case, probably I should have written the original equation I want to solve logarithmically as $a + f_0 + f_1 = f_{01}$, to make it even more clear that I do not expect there to be a unique solution --- multiply the original $F(t_0,q_0,t_1,q_1)$ by $e^{\lambda(t_1 - t_0)}$ for any $\lambda$, for example. But I do expect that there's a "small" family, like is the case for differential equations? –  Theo Johnson-Freyd May 15 '10 at 3:03
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Here's a vague answer then: there's an old 1978 paper in Aequationes Mathematicae mentionning 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 Hamilonian mechanics, I would imagine at least some variant of your problem to have been investigated already. Obviously you're looking at the stationnary phase approximation to the quantum propagator, such as in equation (33.32) of a chapter about the semiclassical propagator in a great physics textbook. Now there's been a lot of mathematical work on the topic which may, at least implicitely, help you with your specific question, namely this paper of Meinrenken in particular page 7 and beyond, and that paper of Combescure-Ralston-Robert.

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