Why foreign physicists' names retain diactrics when the phonetic meaning of these diactrics becomes irrelevant? "Ampère" uses "è" to indicate which type of French "e" to use but English doesn't even differentiate between an open and closed e. In "Schrödinger" the "ö" sound isn't used at all in English. Native dictionaries don't consider these diactrics in ordering words which means their role is merely to indicate pronounciation.
Why not simply write "Ampere" or "Schrodinger"?
This question has little to do with physics and physicists. Rather, it concerns how to treat, in an English text, diacritics and other special characters that appear in non-English names or loanwords. And that, in turn, is a matter of style: different manuals of style will give different recommendations. Thus, your physics sources will (or should) follow the recommendation of whatever style manual they are committed to; I would imagine this will normally be The Chicago Manual of Style/Turabian.
Here is what several style manuals say about the issue of diacritics.
The Chicago Manual of Style
The CMoS recommends using the spelling in the Merriam-Webster dictionary. This raises the question of how that dictionary makes its decisions, and that is unfortunately not entirely clear. For example, we have Göttingen and Västerås, but Łódź apparently was a bridge too far and is rendered as Lodz. (And it wasn't the L with stroke that was the problem there, since the dictionary does use it to render Marie Curie's maiden name, Skłodowska.) As far as the two physicists in your question, they are indeed rendered as Schrödinger and Ampère.
The Associated Press Stylebook
Use accent marks or other diacritical marks with names of people who request them or are widely known to use them, or if quoting directly in a language that uses them: An immigration officer spotted him and asked an innocuous question: “Cómo estás?” How are you? Otherwise, do not use these marks in English-language stories. Note: Many AP customers’ computer systems ingest via the ANPA standard and will not receive diacritical marks published by the AP.
The New York Times Manual of Style and Usage
Accent marks are used for French, Italian, Spanish, Portuguese and German words and names. For simplicity, use the marks uniformly with uppercase and lowercase letters, despite conventions that treat certain uppercase accents as optional. Do not use accents in words or names from other languages (Slavic and Scandinavian ones, for example), which are less familiar to most American writers, editors and readers; such marks would be prone to error.
Some foreign words that enter the English language keep their accent marks (protégé, résumé); others lose them (cafe, facade). The dictionary governs spellings, except for those shown in this manual.
In the name of a United States resident, use or omit accents as the bearer does; when in doubt, omit them. (Exception: Use accents in Spanish names of Puerto Rico residents.)
For more, see this extended discussion: Wikipedia:Naming conventions (use English)/Diacritics RfC.
Finally, note that other people have an opposite impression, namely, that English-language writers are often not retaining the diacritics; see Why does English omit diacritics on foreign names?.
