Resolvent bounds for Lipschitz potentials in dimension two and higher with singularities at the origin
| Autor: | Obovu, Donnell |
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| Rok vydání: | 2023 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | We consider, for $h,E>0$, the semiclassical Schr\"odinger operator $-h^2\Delta + V - E$ in dimension two and higher. The potential $V$, and its radial derivative $\dell_{r}V$ are bounded away from the origin, have long-range decay and $V$ is bounded by $r^{-\delta}$ near the origin while $\dell_{r}V$ is bounded by $r^{-1-\delta}$, where $0\leq\delta\leq 4(\sqrt{2}-1)$. In this setting, we show that the resolvent bound is exponential in $h^{-1}$, while the exterior resolvent bound is linear in $h^{-1}$. Comment: Updates: -Added references. -Revised hypothesis Lemma 2.1. -Revised hypothesis of Lemma 2.2. -Amendments made in proof of Lemma 2.2 to provide extra clarity. -Resolved error made in the calculation of (3.2) -Extra clarification is provided in proofs in Section 4 |
| Databáze: | arXiv |
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