Here are five distinct uses of the word spectrum in physics and mathematics:
- Spectrum (optics): The range of colors in the rainbow
- Spectrum (particle physics): The range of electromagnetic frequencies emitted when an electron in an atom moves from a high energy state to a low energy state
- Spectrum (functional analysis): the set of generalized eigenvalues of a linear map
- Spectrum (algebra): the set of all prime ideals in a ring equipped with the Zariski topology
- Spectrum (topology): a sequence $X_n$ of topological spaces equipped with inclusion maps from the suspension of $X_n$ into $X_{n+1}$
- (Added later): Spectral Sequence (topology): a tool for calculating (co)homology, generalizing exact sequences
My assumption is that these uses of the word spectrum are all related, and I would like to tell a story about how and why the terminology was adapted to different contexts. Here is my speculation:
1 was introduced by Newton in the course of his study of optics. 2 was simply adapted from Newton's use of the word as physicists began to understand the structure of the atom. 3 arose in quantum physics (maybe due to Hilbert or von Neumann?) when it was realized that lines in atomic spectra correspond to eigenvalues of certain linear operators. 4 emerged from the theory of Banach algebras wherein it was realized that if $T$ is a linear operator on Hilbert space which commutes with its adjoint then the maximal ideal space of the commutative C*-algebra generated by $T$ coincides with the spectrum of $T$ in the sense of 3. I have no idea where 5 or 6 comes from (or if they are related).
Can anyone fill in the details of this historical narrative and correct it as necessary? In particular, does anybody know how the word spectrum came to be used in algebraic topology?