Dispersive effects for the Schrödinger equation on a tadpole graph

2015/12/16 by Felix Ali Mehmeti, Kaïs Ammari, Mehmeti, Felix Ali +3 · 1 citation
Mathematics · Physics and Astronomy · #34B45 #34L25 #35B20 #35B40 #47A60 #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.MP #msc:34B45 #msc:34L25 #msc:35B20 #msc:35B40 #msc:47A60

paper · pdf · doi:10.48550/arxiv.1512.05269

arxiv created 2015/12/16 · arxiv updated 2015/12/17

Abstract

We consider the free Schrödinger group e^{-it (d2)/(dx2)} on a tadpole graph \mathcal R. We first show that the time decay estimates L¹ (\mathcal R) → L^∞ (\mathcal R) is in |t|^-\frac12 with a constant independent of the length 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.

Cited by

• Ground States of the Nonlinear Schrödinger Equation on the Tadpole Graph with a Repulsive Delta Vertex Condition

Related

• Dispersive effects and high frequency behaviour for the Schrödinger equation in star-shaped networks
• Dispersive effects for the Schrödinger equation on finite metric graphs with infinite ends
• Spectral analysis and stabilization of the dissipative Schrödinger operator on the tadpole graph
• The nonlinear Schr "odinger equation with white noise dispersion on\n quantum graphs
• Dispersion for the Schrödinger Equation on Networks