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

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_{\rm {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 $C_{{\rm {sfuc}}} > 0$ an $L$-independent constant. The exponential decay condition on $φ$ can alternatively be formulated as an exponential decay condition of the map $λ \mapsto \lVert χ_{[λ, ∞)} (H_L) φ \rVert^2$. 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 $C_{{\rm {sfuc}}}$ in terms of the parameters. Moreover, we show that a similar result cannot hold under a polynomial decay condition.