Dispersive effects for the Schrödinger equation on the tadpole graph

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• 作者：Felix Ali Mehmeti^a ; ^{felix.ali-mehmeti@univ-valenciennes.fr} ; Kaï ; s Ammari^b ; ^{kais.ammari@fsm.rnu.tn} ; Serge Nicaise^a ; ^{snicaise@univ-valenciennes.fr}
• 关键词：Dispersive estimate ; Schrö ; dinger operator ; Tadpole graph
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 April 2017
• 年：2017
• 卷：448
• 期：1
• 页码：262-280
• 全文大小：401 K
• 卷排序：448

文摘

We consider the free Schrödinger group e−itd2dx2 on the tadpole graph RR. We first show that the time decay estimates L1(R)→L∞(R)L1(R)→L∞(R) is in |t|−12 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.