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In various articles (I am here talking about specially the ones related to string theory etc.) I have seen the discussion on density and distribution of eigenvalues. I want to know why do we use them (why are they important to consider in some calculation, how to construct them), if there's any physical significances behind them etc. And anything else that you might want to mention. And again a reference could be useful. Thanks!

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Well, whenever there is a large number of "something", it is always important and relevant to study the "something" by statistical methods. For example, there are many people in China, so it's sensible to ask how many people are there in China, and what is their population density - in the whole China and in its parts.

The density of states $\rho(E)$ whose energy is near a certain value determines the entropy at this value of energy, $S\sim \ln(\rho(E))$: $S$ is typically large so the density of states has to be exponentially large. In string theory, such functions may usually be calculated in several very different ways and compared. The density of high-energy states is, for example, related to the energy of the ground state (by "modular invariance").

Just like the population density as a function of the latitude and longitude is relevant, so is the whole distribution of the eigenvalues. It may be approximated by various approximate formulae.

It's also important to know how the spacing between two eigenvalues is distributed relatively to the other spacing. If the eigenvalues were numbers that determine the positions of randomly and independently falling needles, the distribution would be "Poisson": the eigenvalues as functions of energy would display the same patterns as radioactive beeps of a counter as a function of time.

However, in quantum systems, the eigenvalues are often distributed differently. Very often, the distribution is similar to that of "random matrix theory" in which the eigenvalues are much less likely to be very close to each other - we say that the "eigenvalues repel". So the beeping would be much more "rhythmical" than the beeping of the radioactive counters. Ironically enough, the energy levels of large atomic nuclei are the most mundane example of a physical system whose eigenvalues repel - and therefore differ from the beeping of the radioactive counters. ;-)

All such statistical tests are doable and they tell us something. So they're important. They're used in every other page of a paper that studies a system with a large number of eigenvalues or eigenstates - which includes any large enough system. One can't enumerate all the situations in which such information is relevant much like one can't give you the references to all papers that use the population density in China.

For some introduction to eigenvalues, I may recommend you Jake Barnett, a 12-year-old student of Purdue University in Indianopolis. Here is his tutorial on eigenvalues:

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