通过库仑碰撞的半经典测度
Semiclassical measures through Coulomb collisions
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中文总结 AI 辅助
研究与吸引库仑算子能量E<0的本征函数序列相关的半经典测度，通过Moser - Fock映射及技术算子扩展引理，证明μ是半经典测度的充要条件，解决了Keraani提出的开放问题。
中文摘要 AI 辅助
我们证明，当且仅当μ是能量（超）曲面Σ_E上在Moser正则化开普勒流作用下不变的概率测度时，μ是与吸引库仑算子能量E<0的本征函数序列相关的半经典测度。作者近期工作已证明了逆命题，本文证明了另一个方向（以及逆命题的独立证明）。我们针对允许在无穷远处有一定非衰减的一般符号类证明了主要定理，这意味着半经典测度质量完全从原点反射。在精确库仑算子本征函数序列的半经典测度的特殊情况下，本文解决了Keraani提出的一个开放问题。主要工具包括著名的Moser - Fock映射以及\(\Psi_{\hbar}^0(\mathbb{S}^d)\)中的一个技术算子扩展引理，该引理利用了标准本征函数集中界。
英文摘要
We prove that $μ$ is a semiclassical measure associated to a sequence of eigenfunctions of energy $E<0$ of the attractive Coulomb operator if and only if $μ$ is a probability measure on the energy (hyper)surface $Σ_E$ invariant under the regularized Kepler flow due to Moser. The converse was shown in recent work by the author, and the present article proves the other direction (as well as an independent proof of the converse). We prove the main theorem for a general symbol class allowing certain non-decay at infinity, which implies that semiclassical measure mass entirely reflects off of the origin. In the special case of semiclassical measures of sequences of eigenfunctions of the exact Coulomb operator, this article solves an open problem posed by Keraani. The main tools include the celebrated Moser-Fock map along with a technical operator extension lemma in $Ψ_{\hbar}^0(\mathbb{S}^d)$, which utilizes standard eigenfunction concentration bounds.
