On the ergodic limit of the spectral shift function
by
Vita Borovyk
University of Missouri-Columbia
Coauthors: Konstantin Makarov

We study Schrödinger operators on [0, ∞) (with the Dirichlet boundary condition at zero) using box approximation. We establish a connection between the characteristics of the absolutely continuous spectrum of the half-line problem and the discrete spectrum of the finite interval problem. More precisely, we prove that

lim R→ ∞  p R ó õ R 0  æ è N0r(l) -Nr(l) ö ø dr = d æ è Ö   l   ö ø ,

where N₀^r is the counting function for the free Schrödinger operator on the interval [0, r], N^r is the counting function for the operator with potential, and d is the phase shift associated with the half-line problem.

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Date received: February 13, 2008