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 ${\Lambda }_L=(-L/2,L/2{)}^d$ and $H_L=-{\Delta }_L+V_L$ be a Schrödinger operator on $L^2({\Lambda }_L)$ with a bounded potential $V_L:{\Lambda }_L\to {\mathbb{R}}^d$ 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 $\varphi$ is an infinite complex linear combination of eigenfunctions of $H_L$ with exponentially decaying coefficients, $W_{\delta }(L)$ is some union of equidistributed $\delta$-balls in ${\Lambda }_L$ and $C_{\mathrm{sfuc}}>0$ an $L$-independent constant. The exponential decay condition on $\varphi$ can alternatively be formulated as an exponential decay condition of the map $\lambda \mapsto \Vert {\chi }_{[\lambda ,\infty )}(H_L)\varphi {\Vert }^2$. The novelty is that at the same time we allow the function $\varphi$ 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.

References in corpus (2)

• Scale-free unique continuation principle, eigenvalue lifting and Wegner estimates for random Schrödinger operators
• Scale-free uncertainty principles and Wegner estimates for random breather potentials

Cited by in corpus (2)

• Scale-free unique continuation principle, eigenvalue lifting and Wegner estimates for random Schrödinger operators
• Scale-free quantitative unique continuation and equidistribution estimates for solutions of elliptic differential equations