Is there any sensible meaning of the term eigenfunctionals? The object I want to describe is a solution to the following equation
$$ {\mathscr D}_x F[g] = f(x) F[g] $$
where $ {\mathscr D}_x$ is an operator that maps functionals to functionals (e.g. a functional derivative $\frac{\delta}{\delta g(x)}$), $F$ is an eigenfunctional of ${\mathscr D}_x$, $f(x)$ is a corresponding eigenfunction, and $g(x)$ is some other function. Of course, this is rather a try to make sense of this term or to motivate thinking about it than a proper definition since I cannot find any. Additionally, is there an application of this concept in physics, e.g. when using functional methods in the path integral formalism of quantum field theory? A special case I encountered during studying correlation functions would be:
$$F[\phi(x)] = \exp \left( \int dx \, \phi(x) \, h(x) \right) $$
$$\frac{\delta F[\phi(x)]}{\delta \phi(y)} = h(y) F[\phi(x)]$$
However, a generalization of the idea would probably be fruitful.
