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Eigenvalue distribution of random operators and matrices. (English) Zbl 0790.35081

Séminaire Bourbaki, Vol. 1991/92. Exposés 745-759 (avec table par noms d’auteurs de 1948/49 à 1991/92). Paris: Société Mathématique de France, Astérisque. 206, 445-461 (Exp. No. 758) (1992).
According to H. Weyl the so-called counting function for an elliptic operator \(A_ \Lambda= -\Delta\) has the form: \[ {\mathcal N}_ \Lambda (\lambda)= c_ d|\Lambda| \lambda^{d/2}+ o(\lambda^{d/2}), \] where \(\Lambda\) is a compact domain in \(\mathbb{R}^ d\), \(|\Lambda|\) is the volume of \(\Lambda\), \(\lambda\) denotes an eigenvalue, \(c_ d\) is a constant dependent on the dimension of the space \(\mathbb{R}^ d\).
When \(\Lambda_ k\) is a sequence of compact domains expanding into the whole \(\mathbb{R}^ d\) as \(k\to\infty\) then it is interesting to find the limit: \[ \lim_{k\to\infty} N_{\Lambda_ k} (\lambda)\equiv N(\lambda), \quad \text{ where } \quad N_ \Lambda(\lambda)= |\Lambda|^{-1} {\mathcal N}_ \Lambda(\lambda). \] The problem of studying of \(N_ \Lambda(\lambda)\) is known as the integrated density of states (IDS).
In the paper the problem of studying the IDS for three classes of random differential and matrix operators is discussed. The classes are the following: 1) differential and finite-difference operators with random coefficients, 2) random matrices with independent and identically distributed entries, 3) this class includes operators that are in a certain sense interpolating between 1) and 2).
The paper contains a survey of the results with references on the mentioned problem and also recalls their physical motivations.
For the entire collection see [Zbl 0772.00016].

MSC:

35P20 Asymptotic distributions of eigenvalues in context of PDEs
35R60 PDEs with randomness, stochastic partial differential equations
15B52 Random matrices (algebraic aspects)
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
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