The Variational Structure of Riesz Means
Year: 2026
Type: preprint
Abstract: We establish the variational structure of Riesz means of negative eigenvalues of Schrödinger operators. For Hₕ(y) = −h²Δ − y(x) on L²(ℝᵈ), with negative eigenvalues λₙ = −κₙ² and L²-normalised eigenfunctions ψₙ, the Hellmann–Feynman theorem and the chain rule yield, for every Riesz order γ > ½, the functional identityδ/δy(x) ∑ₙ κₙ^(2γ) = γ ∑ₙ κₙ^(2γ−2) ψₙ(x)²,for h > 0, without recourse to a semiclassical limit.At the distinguished order γ = 3/2, the identity specialises to δSₕ/δy = (3/2)∑ₙ κₙψₙ², with Sₕ = ∑ₙ κₙ³; in dimension one this realises the reconstructionyₕ(x) := 4h∑ₙ κₙψₙ(x)² = (8h/3) δSₕ/δy(x),as the exact functional gradient of the Riesz mean R₃⁄₂—the (previously postulated) Semi-Classical Signal Analysis formula, here derived rather than assumed. Combined with the sharp Lieb–Thirring inequalitySₕ = ∑ₙ κₙ³ ≤ L₃⁄₂,ᵈᶜˡ h⁻ᵈ ∫ y^(3/2+d/2),proved in all dimensions by Laptev and Weidl, and with the Weyl asymptotics, three consequences follow in dimension one. First, yₕ decomposes as yₕ = y + (8h/3) δℛₕ/δy, where ℛₕ is the Weyl remainder, with ‖y − yₕ‖L² = O(h). Second, an error-gap identity y − yₕ = (8h/3) δΔₕ/δy holds, where Δₕ ≥ 0 is the Lieb–Thirring gap. Third, perfect reconstruction is characterised as saturation of the Lieb–Thirring bound.The order γ = 3/2 is uniquely determined: it is the only Riesz order whose functional gradient recovers a linear reconstruction formula weighted by κₙψₙ². The framework extends to all dimensions d ≥ 1, with linearity order γₗᵢₙ(d) = 2 − d/2; dimension d = 1 is the only case in which linearity, the classical constant, and established error estimates coincide.
Source: HAL
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7165437201