Control for Schrödinger operators on 2-tori: rough potentials

Bourgain, Jean, Burq, Nicolas, and Zworski, Maciej. "Control for Schrödinger operators on 2-tori: rough potentials." Journal of the European Mathematical Society 015.5 (2013): 1597-1628. <http://eudml.org/doc/277754>.

@article{Bourgain2013,
abstract = {For the Schrödinger equation, $(i\partial _t+\nabla )u=0$ on a torus, an arbitrary non-empty open set $\Omega $ provides control and observability of the solution: $\left\Vert u\left|_\{t=0\}\right\Vert _\{L^2(\mathbb \{T\}^2)\}\le K_T\left\Vert u\right\Vert _\{L^2([0,T]\times \Omega )\}\right.$. We show that the same result remains true for $(i\partial _t+\nabla -V)u=0$ where $V\in L^2(\mathbb \{T\}^2)$, and $\mathbb \{T\}^2$ is a (rational or irrational) torus. That extends the results of [1], and [8] where the observability was proved for $V\in C(\mathbb \{T\}^2)$ and conjectured for $V\in L^\{\infty \}(\mathbb \{T\}^2)$. The higher dimensional generalization remains open for $V\in L^\{\infty \}(\mathbb \{T\}^n)$.},
author = {Bourgain, Jean, Burq, Nicolas, Zworski, Maciej},
journal = {Journal of the European Mathematical Society},
keywords = {Schrödinger equation; control; observability; two-dimensional tori; Schrödinger equation; control; observability; two-dimensional tori},
language = {eng},
number = {5},
pages = {1597-1628},
publisher = {European Mathematical Society Publishing House},
title = {Control for Schrödinger operators on 2-tori: rough potentials},
url = {http://eudml.org/doc/277754},
volume = {015},
year = {2013},
}

TY - JOUR
AU - Bourgain, Jean
AU - Burq, Nicolas
AU - Zworski, Maciej
TI - Control for Schrödinger operators on 2-tori: rough potentials
JO - Journal of the European Mathematical Society
PY - 2013
PB - European Mathematical Society Publishing House
VL - 015
IS - 5
SP - 1597
EP - 1628
AB - For the Schrödinger equation, $(i\partial _t+\nabla )u=0$ on a torus, an arbitrary non-empty open set $\Omega $ provides control and observability of the solution: $\left\Vert u\left|_{t=0}\right\Vert _{L^2(\mathbb {T}^2)}\le K_T\left\Vert u\right\Vert _{L^2([0,T]\times \Omega )}\right.$. We show that the same result remains true for $(i\partial _t+\nabla -V)u=0$ where $V\in L^2(\mathbb {T}^2)$, and $\mathbb {T}^2$ is a (rational or irrational) torus. That extends the results of [1], and [8] where the observability was proved for $V\in C(\mathbb {T}^2)$ and conjectured for $V\in L^{\infty }(\mathbb {T}^2)$. The higher dimensional generalization remains open for $V\in L^{\infty }(\mathbb {T}^n)$.
LA - eng
KW - Schrödinger equation; control; observability; two-dimensional tori; Schrödinger equation; control; observability; two-dimensional tori
UR - http://eudml.org/doc/277754
ER -