Application of the spectrum of an operator http://en.wikipedia.org/wiki/Spectrum_of_an_operator
What is the application of the spectrum of an operator
 A: If you describe the behaviour of an electron in the hydrogren atom with the help of a Hamiltoninan Operator, then you end up with a differential operator (first in $L^2(\mathbb{R^3})$, but you can use some symmetry properties to end up with a Sturm-Liouville operator in $L^2(\mathbb{R^+})$). This hamiltonian operator has a spectrum which is bounded from below, has a number of discrete eigenvalues which accumulate towards the continuous spectrum and an unbounded continuous spectrum. The interesting part of this story is: This is what the absorption spectrum of Hydrogen looks like (actually, only the Lyman Series). The linked picture displays more absorption series, which can also be accounted with by modifying the hamiltonian accordingly.
This precision is one of the reasons why quantum mechanics and operator theory became a successful tool in the early 20th century: Have a good description of the absorption behaviour of atoms (i.e. precise knowledge about their energy states).
A: To paraphrase a famous slogan, "all mathematics is operator theory".  This is even more true of applicable mathematics.  I would guess that eigenvalue and spectral calculations account for a significant fraction of the world's CPU time spent on computation.
