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In Methods of Mathematical Physics IV by Reed and Simon, the authors cover Floquet theory in detail in Section XIII.16. On page 280, they note that

"A part of the analysis of [the periodic Schrödinger operator] is then a special case of general symmetry arguments which are the subject of Chapter XVI."

Unfortunately, Chapter XVI, which was to be included in a future volume under the name "Introduction to Group Representations," was never actually written. Does anyone know of a good resource covering group representations specifically in the context of spectral theory of Schrödinger operators (ie, for analysts rather than algebraists)?

The closest I've seen are books with titles like "Quantum mechanics for mathematicians" (such as those by Hall, Teschl, Takhtajan) with chapters on angular momentum, but they don't really cover it from the perspective of spectral theory, and don't treat everything in a unified context (for instance, none of those books mention Floquet theory).

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There are the Books and Papers by Kuchment e.g. "An overview of periodic elliptic operators" or, if you're more into the manifold setting there are some papers by Sunada (e.g. "Periodic Schrödinger Operators on a Manifold"). The case of non abelian symmetry groups is dealt with in the PhD thesis of Michael Gruber, a survey paper would be "Non-commutative Bloch theory. An Overview".

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