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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.

Qmechanic
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Lentes
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    I think it depends on a particular physical problem, since SE is a rather general equation. In continuous spectrum dnesity-of-states determines many physical properties - that is the number of eigenvalues per energy interval. – Roger V. Sep 13 '21 at 09:41
  • Thank you for your comment! I would appreciate some very introductory remarks about how the number of eigenvalues (i.e. the integrated density of states) helps determine physical properties (for instance, which physical properties? etc.).

    I am interested both in the case of continuous spectrum and in the case of discrete spectrum.

     – Lentes Sep 13 '21 at 10:36
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    I wrote a short answer to give you a better idea. It intends by no means to be comprehensive. My field is electron transport, so this is where I place the emphasis, but DOS is ubiquitous in physics whenever a continuous spectrum arises. – Roger V. Sep 13 '21 at 11:34

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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:

Roger V.
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  • Thanks for your answer! I will soon check the links thoroughly. In the meantime, I will leave the question open, in case other people would like to mention other applications. – Lentes Sep 15 '21 at 21:58