Though this does not directly answer your question, here is a foundational paper that might help one derive results that might answer your question:
Marian Boykan Pour-El and Ian Richards: "Noncomputability in Analysis and Physics: A Complete Determination of the Class of Noncomputable Linear Operators", Advances in Mathematics 48, 44-74 (1983).
I quote the short first paragraph of this paper as it sets the tone for what follows:
"One would assume that a "reasonable" operator should map computable input data onto computable solutions. It is perhaps surprising that many of the standard operators of analysis and physics fail to do this. In this article, we shall determine precisely which linear operators do, and which do not, preserve computability."
I hope this paper helps.
Addendum: Consider their Main Theorem and its Complement:
Main Theorem: Let $X$ and $Y$ be Banach spaces with computability theories, and let $e_{n}$ be an effective generating set for $X$. Let $T$: $X$$\rightarrow$$Y$ be a closed linear operator whose domain includes {$e_{n}$} and such that $T$$e_{n}$ is a computable sequence in $Y$. Then $T$ maps every computable element of its domain onto a computable element of $Y$ if and only if $T$ is bounded.
Complement. Under the same assumptions, if $T$ is bounded then more can be said. The domain of $T$ coincides with $X$, and $T$ maps every computable sequence in $X$ onto a computable sequence in $Y$.
Working backward to discover the philosophical motivation for their first paragraph, it seems the place to start in order to analyze the non-computabilty in analysis and physics they seek to show.
Does this help any, Morteza?