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Oscillation theory for the density of states of high dimensional random operators. (English) Zbl 1459.81039

Int. Math. Res. Not. 2019, No. 15, 4579-4602 (2019); erratum ibid. 2019, No. 15, 4898 (2019).
Summary: Sturm-Liouville oscillation theory is studied for Jacobi operators with block entries given by covariant operators on an infinite dimensional Hilbert space. It is shown that the integrated density of states of the Jacobi operator is approximated by the winding of the Prüfer phase w.r.t. the trace per unit volume. This rotation number can be interpreted as a spectral flow in a von Neumann algebra with finite trace.

MSC:

81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
34B24 Sturm-Liouville theory
81R20 Covariant wave equations in quantum theory, relativistic quantum mechanics
47B36 Jacobi (tridiagonal) operators (matrices) and generalizations
58J30 Spectral flows
46L10 General theory of von Neumann algebras
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
58J22 Exotic index theories on manifolds
82B30 Statistical thermodynamics
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