Szegö limit theorems in quantum mechanics

Steven Zelditch*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


We prove a Szegö-type theorem for some Schrödinger operators of the form H = -1 2Δ + V with V smooth, positive and growing like V0|x|k, k > 0. Namely, let πλ be the orthogonal projection of L2 onto the space of the eigenfunctions of H with eigenvalue ≤λ; let A be a 0th order self-adjoint pseudo-differential operator relative to Beals-Fefferman weights θ{symbol}(x, ξ) = 1, Φ(x, ξ) = (1 + |ξ|2 + V(x)) 1 2 and with total symbol a(x, ξ); and let f∈C(R). Then lim λ→∞ 1 rankπλtrf(πλλ)= lim λ→∞ 1 vol(H(x, ξ)≤λ) ∫ H≤λf(a(x, ξ))dxdξ (assuming one limit exists).

Original languageEnglish (US)
Pages (from-to)67-80
Number of pages14
JournalJournal of Functional Analysis
Issue number1
StatePublished - Jan 1983

ASJC Scopus subject areas

  • Analysis


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