Decay estimates for one Aharonov-Bohm solenoid in a uniform magnetic field II: wave equation
| Autor: | Wang, Haoran, Zhang, Fang, Zhang, Junyong |
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| Rok vydání: | 2023 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | This is the second of a series of papers in which we investigate the decay estimates for dispersive equations with Aharonov-Bohm solenoids in a uniform magnetic field. In our first starting paper \cite{WZZ}, we have studied the Strichartz estimates for Schr\"odinger equation with one Aharonov-Bohm solenoid in a uniform magnetic field. The wave equation in this setting becomes more delicate since a difficulty is raised from the square root of the eigenvalue of the Schr\"odinger operator $H_{\alpha, B_0}$ so that we cannot directly construct the half-wave propagator. An independent interesting result concerning the Gaussian upper bounds of the heat kernel is proved by using two different methods. The first one is based on establishing Davies-Gaffney inequality in this setting and the second one is straightforward to construct the heat kernel (which efficiently captures the magnetic effects) based on the Schulman-Sunada formula. As byproducts, we prove optimal bounds for the heat kernel and show the Bernstein inequality and the square function inequality for Schr\"odinger operator with one Aharonov-Bohm solenoid in a uniform magnetic field. Comment: 35 pages, comments are welcome! |
| Databáze: | arXiv |
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