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.

  • $\begingroup$ This is a Schwinger-Dyson equation. See this PSE post for the standard use in physics. $\endgroup$ – AccidentalFourierTransform Oct 23 '19 at 23:24
  • $\begingroup$ In the Schrödinger picture of a quantum field theory, one has wavefunctionals of the fields which are eigenfunctionals of the Hamiltonian. Here is a nice introduction: sciencedirect.com/science/article/pii/055032138590210X $\endgroup$ – Seth Whitsitt Oct 23 '19 at 23:31
  • $\begingroup$ Thank you, accidentalfouriertransform, for the reference to the great post, which will be helpful in understanding the Schwinger-Dyson equation! However, I rather intended to get some insights into the mathematical properties of the more general equation I proposed. Do you know of another application of this kind of equation? $\endgroup$ – Carlo Tasillo Oct 23 '19 at 23:34
  • $\begingroup$ @SethWhitsitt: Thanks for pointing out the connection to the Schrödinger picture of QFT which I haven't heard of before. I will check if I will be able to find the mathematical foundation for its defining equation which should be equivalent to my question. $\endgroup$ – Carlo Tasillo Oct 23 '19 at 23:48
  • $\begingroup$ @CarloTasillo My answers to the questions physics.stackexchange.com/q/441251 and physics.stackexchange.com/q/439434 both mention this kind of formulation (Schrodinger functional formulation of QFT). $\endgroup$ – Chiral Anomaly Oct 24 '19 at 13:06

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