Why is the multiset of eigenvalues called spectrum?

Solution 1:

Not too surprisingly, it seems to come from physics. I'm sure others here can provide much better detail, but here is a quote from a small article called "Favorite Eigenvalue Problems" in SIAM News, Volume 44, Number 10, December 2011 by Nick Trefethen:

Eigenvalues played a role in a great coincidence of scientific history. Physicists saw spectral lines in the light from stars; indeed, an unfamiliar line led to the discovery of helium. Later, Hilbert defined spectra of operators. Not until Schrödinger, decades on, was it understood that physicists’ spectra were exactly a case of mathematicians’ spectra, with each line corresponding to the difference in energy of two eigenstates. And it was these same spectral lines that led to the discovery of the red shift and the expanding universe.

Solution 2:

Newton introduced the word "spectrum" in sciences from 1666. The terminology of the eigen-elements of a matrix is fixed by Jordan in 1870 (in fact, for a while, mathematicians use "proper value" (Jordan) or "eigenvalue" (Hilbert)).

In 1907, functional analysis appears; in 1910, Weyl gives his definition of what would later be called a variant of the "essential spectrum".

Then comes the equivalence between the Heisenberg's matricial mechanics and the Schrodinger's equation (1926). Yet, the word spectrum still does not exist in the sense that interests us.

Note that a spectrum (for its future standard definition) of a Banach operator may be discrete or continuous (as $\phi:f(x)\in C^0([0,1])\rightarrow xf(x)\in C^0([0,1]))$. Therefore, the so called spectrum has some points in common with the spectrum of physicists. Moreover these two notions give each one a lot of information about the studied object.

Anyway, in english, "spectrum" is used -for operators- from 1948. Since in finite dimension, the spectrum reduces to the set of eigenvalues, the word "spectre" is used in France -for the matrices- from 1964; on the other hand, "spectrum" is pronounced faster than "the set of eigenvalues"!!