Scale-free and quantitative unique continuation for infinite dimensional spectral subspaces of Schrödinger operators

Abstract

We prove a quantitative unique continuation principle for infinite dimensional spectral subspaces of Schrödinger operators. Let $Λ_L = (-L/2,L/2)^d$ and $H_L = -Δ_L + V_L$ be a Schrödinger operator on $L^2 (Λ_L)$ with a bounded potential $V_L : Λ_L \to \mathbb{R}^d$ and Dirichlet, Neumann, or periodic boundary conditions. Our main result is of the type \[ \int_{Λ_L} \lvert ϕ\rvert^2 \leq C_{\mathrm{sfuc}} \int_{W_δ(L)} \lvert ϕ\rvert^2, \] where $ϕ$ is an infinite complex linear combination of eigenfunctions of $H_L$ with exponentially decaying coefficients, $W_δ(L)$ is some union of equidistributed $δ$-balls in $Λ_L$ and $C_{\mathrm{sfuc}} > 0$ an $L$-independent constant. The exponential decay condition on $ϕ$ can alternatively be formulated as an exponential decay condition of the map $λ\mapsto \lVert χ_{[λ, \infty)} (H_L) ϕ\rVert^2$. The novelty is that at the same time we allow the function $ϕ$ to be from an infinite dimensional spectral subspace and keep an explicit control over the constant $C_{\mathrm{sfuc}}$ in terms of the parameters. Moreover, we show that a similar result cannot hold under a polynomial decay condition.