Dispersive effects for the Schr\"odinger equation on a tadpole graph
We consider the free Schr\"odinger group $e^{-it \frac{d^2}{dx^2}}$ on a tadpole graph ${\mathcal R}$. We first show that the time decay estimates $L^1 ({\mathcal R}) \rightarrow L^\infty ({\mathcal R})$ is in $|t|^{-\frac12}$ with a constant independent of the length of the circle. Our proof i...
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| Main Authors | , , |
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| Format | Journal Article |
| Language | English |
| Published |
16.12.2015
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| Subjects | |
| Online Access | Get full text |
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| Summary: | We consider the free Schr\"odinger group $e^{-it \frac{d^2}{dx^2}}$ on a
tadpole graph ${\mathcal R}$. We first show that the time decay estimates $L^1
({\mathcal R}) \rightarrow L^\infty ({\mathcal R})$ 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. |
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| DOI: | 10.48550/arxiv.1512.05269 |