Szego limit theorem on the lattice

2011/02/21 by Swain, Jitendriya, Krishna, M.
#FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.1102.4131

Abstract

In this paper, we prove a Szegö type limit theorem on ℓ²(\ZZ^d). We consider operators of the form H=Δ+V, V multiplication by a positive sequence \V(n), n ∈ \ZZ^d\ with V(n) → ∞, |n| → ∞ on ℓ²(\ZZ^d) and π_λ the orthogonal projection of ℓ²(ℤ^d) on to the space of eigenfunctions of H with eigenvalues ≤ λ. We take B to be a pseudo difference operator of order zero with symbol b(x,n), (x,n) ∈ \TT^d× \ZZ^d and show that for nice functions f lim_{λ→ ∞} Tr(f(π_λBπ_λ))/Tr(π_λ) = lim_{λ→ ∞} (1)/((2π)^d) \frac∑_{V(n) ≤ λ} ∫_\TT^d f(b(x,n)) ~ dx∑_V(n)≤λ 1.

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