# Control for Schrödinger operators on 2-tori: rough potentials

Jean Bourgain; Nicolas Burq; Maciej Zworski

Journal of the European Mathematical Society (2013)

- Volume: 015, Issue: 5, page 1597-1628
- ISSN: 1435-9855

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topBourgain, 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 -

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