Optimal Covariance Estimates for Schrödinger Semigroups with White Noise in d=1,2

2026/07/19 by Youssef Djellouli, Pierre Yves Gaudreau Lamarre
#math.PR #math-ph #math.MP

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Abstract

For d∈\1,2\, let H=-(1)/(2)Δ+ V +ξ be the random Schrödinger operator on L²(ℝ^d) where ξ is a standard Gaussian white noise and V is a deterministic potential with power-law growth at infinity. Using a Feynman-Kac formula for the trace of the Schrödinger semigroup, we give optimal asymptotic upper and lower bounds on the covariance of Tr[e^-sH] and Tr[e^-tH] as s,t→0 through estimates on Brownian bridge local times. These estimates are a significant improvement on previous bounds in the case d=1 and are the first of their kind for d=2. As an application of these new estimates, we prove a quantitative hyperuniformity-type property and decorrelation rate for the trace as s,t→0.

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