# American Institute of Mathematical Sciences

September  2017, 16(5): 1719-1730. doi: 10.3934/cpaa.2017083

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

 1 Technische Universität Dortmund, Fakultät für Mathematik, 44227 Dortmund, Germany 2 Technische Universität Chemnitz, Fakultät für Mathematik, 09126 Chemnitz, Germany

* Corresponding author

Received  September 2016 Revised  February 2017 Published  May 2017

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.
Citation: Matthias Täufer, Martin Tautenhahn. Scale-free and quantitative unique continuation for infinite dimensional spectral subspaces of Schrödinger operators. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1719-1730. doi: 10.3934/cpaa.2017083
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