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Bounded Sobolev norms for linear Schrödinger equations under resonant perturbations. (English) Zbl 1171.35029

Consider time periodic Schrödinger equations \[ i\frac{\partial u}{\partial t}=-\Delta u+V(x,t)u (1) \] with periodic boundary conditions, where \(x\in \mathbb T\), \(t\in \mathbb R\), \(V\) is a real analytic potential periodic in \(x\) and \(t\). Let us assume that the frequncy \(\omega\) of this time periodic potential \(V\) is an integer, \(\omega\in\mathbb Z \setminus \{0\}\) (the case of resonant perturbation). Let \(H^s(\mathbb T )\), \(s>0\), be a Sobolev space on \(\mathbb T\), \(\| \cdot\|_s\) the norm in \(H^s(\mathbb T )\), \(u(t)\) the solution of (1) with initial condition \(u(0)\in H^s(\mathbb T )\). Under certain conditions the author proves that for all \(s>0\), the solution \(u(t)\) satisfies \[ \|u(t)\|_s\leq C_s\|u(0)\|_s (2) \] for all \(t\), where \(C_s\) is some positive constant depending only on \(s\). He shows that these conditions are verified for the potential \(V(x,t)=2\cos x\cos t\) and for small resonant perturbations; the inequality (2) holds for such potentials \(V\).

MSC:

35J10 Schrödinger operator, Schrödinger equation
35B20 Perturbations in context of PDEs
35B45 A priori estimates in context of PDEs
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References:

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