Dispersive effects for the Schrodinger equation on the tadpole graph

Abstract : We consider the free Schrodinger group e(-itd2/dx2) on the tadpole graph R. We first show that the time decay estimates L-1(R) -> L infinity(R) is in vertical bar t vertical bar(-1/2) with a constant independent of the circumference of the circle. Our proof is based on an appropriate decomposition of the kernel of the resolvent. Further we derive a dispersive perturbation estimate, which proves that the solution on the queue of the tadpole converges uniformly, after compensation of the underlying time decay, to the solution of the Neumann half-line problem, as the circle shrinks to a point. To obtain this result, we suppose that the initial condition fulfills a high frequency cutoff. (C) 2016 Elsevier Inc. All rights reserved.
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Felix Ali Mehmeti, Kaïs Ammari, Serge Nicaise. Dispersive effects for the Schrodinger equation on the tadpole graph. Journal of Mathematical Analysis and Applications, Elsevier, 2017, 448 (1), pp.262-280. ⟨10.1016/j.jmaa.2016.10.060⟩. ⟨hal-02152586⟩



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