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Control for Schrödinger operators on tori. (English) Zbl 1281.35011
It is known that solutions of the equation \[ (-\Delta - \lambda)u(z) = f(z) \tag{1} \] on the torus \(\mathbb{T}^2 = \mathbb{R}^2 / A\mathbb{Z} \times B\mathbb{Z}\), where \(A,B \in \mathbb{R}\setminus\{0\}\), can be controlled by an arbitrary non-empty open set \(\Omega \subset \mathbb{T}^2\) in the sense that there exists a constant \(K = K(\Omega)\), depending only on \(\Omega\), such that the \(L^2\)-estimate \[ \|u\|_{L^2(\mathbb{T})} \leq K(\|f\|_{L^2(\mathbb{T})} + \|u\|_{L^2(\Omega)}) \tag{2} \] holds for any \(f \in L^2(\mathbb{T})\) and any corresponding solution \(u\) of (1) (see, S. Jaffard [Port. Math. 47, No. 4, 423–429 (1990; Zbl 0718.49026)]). The authors show in the current paper that this result extends unchanged the Schrödinger equation \[ (-\Delta + V(z) - \lambda)u(z) = f(z), \] \(z \in \mathbb{T}^2\), where significantly the constant in (2) remains independent of the (real-valued) potential \(V \in C^\infty (\mathbb{T}^2)\). A similar estimate is shown to hold for the dynamical Schrödinger equation \[ \mathrm{i} \partial_t u(t,z) = (-\Delta + V(z))u(t,z), \tag{3} \] \(z \in \mathbb{T}^2\). More precisely, for any non-empty open set \(\Omega \subset \mathbb{T}^2\) and any time \(T>0\) there exists a constant \(K=K(\Omega,T)\) such that \[ \|u(0,\,.\,)\|_{L^2(\mathbb{T})}^2 \leq K \int_0^T \|u(t,\,.\,)\|_{L^2(\mathbb{T})}^2\,dt \tag{4} \] for any solution \(u\) of (3).
The proofs are based on methods used in earlier works of the authors to study (1) for high energy frequencies \(\lambda \to \infty\) and involve proving a semi-classically localized version of (4), the latter together with the Duhamel formula implying (2).

MSC:
35J10 Schrödinger operator, Schrödinger equation
35Q41 Time-dependent Schrödinger equations and Dirac equations
35P99 Spectral theory and eigenvalue problems for partial differential equations
58J05 Elliptic equations on manifolds, general theory
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
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