Why are eigenvalue asymptotics (or more generally, the counting of eigenvalues below a certain energy) important to physics?

Question

I am a mathematician, and I have worked with time-independent Schrodinger operators; but I have little rigorous knowledge of classical mechanics or quantum mechanics. I know what eigenvalues and eigenfunctions are from the mathematical viewpoint, and I have obtained some results regarding bounds of eigenvalue counting functions. Out of curiosity, I would appreciate a physicist's perspective on why eigenvalue asymptotics (for instance, Weyl's law and related asymptotics for time-independent Schrodinger operators), or more generally the counting of eigenvalues, are interesting things to know in their field.

Answer 1

As I pointed in the comments, one possible application is calculations of the density-of-states (DOS), which determines many physical properties in various contexts:

• absorption of crystals (DOS of final states)
• thermodynamic properties of materials (e.g., via the DOS of phonons)
• electric conductance
• lifetimes of atomic levels
• critical parameters of phase transitions (e.g., metal-insulator transition)

and others. It would be probably easiest to browse the search results in this site, although most of the results will deal with very simple forms of Schrödinger operator (which usually appears via the dispersion relation). Just to give a few links to my own answers:

• electronic transport and quantum Hall effect
• DOS for simple spectra of phonons and free electrons
• DOS in thermodynamic calculations
• DOS in expression for tunneling rate
• generalizations of DOS for interacting spectrum with application to electron transport