The Spectrum of Riemannium Erbium and Eigenvalium Among the
spectra in Figure 1 is a series of 100 energy levels of an
atomic nucleus, measured 30 years ago with great finesse by H.
I. Liou and James Rainwater and their colleagues at Columbia
University. The nucleus in question is that of the rare-earth
element erbium-166. A glance at the spectrum reveals no obvious
patterns; nevertheless, the texture is quite different from that
of a purely random distribution. In particular, the erbium
spectrum has fewer closely spaced levels than a random sequence
would. It's as if the nuclear energy levels come equipped with
springs to keep them apart. This phenomenon of "level repulsion"
is characteristic of all heavy nuclei. What kind of mathematical
structure could account for such a spectrum? This is where those
eigenvalues of random Hermitian matrices enter the picture. They
were proposed for this purpose in the 1950s by the physicist
Eugene P. Wigner. As it happens, Wigner was another
Princetonian, who could therefo