CPAA
Scale-free and quantitative unique continuation for infinite dimensional spectral subspaces of Schrödinger operators
Matthias Täufer Martin Tautenhahn
Communications on Pure & Applied Analysis 2017, 16(5): 1719-1730 doi: 10.3934/cpaa.2017083
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.
keywords: Unique continuation uncertainty principle Schrödinger operator observability estimate multi-scale domain

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