可积系统准模的非集中性
Non-concentration of quasimodes for integrable systems
摘要：
我们考虑一个算子的本征函数序列（或者更一般地说，一个准模）在相空间中的可能的浓度，该算子的主象征具有完全可积的Hamilton流。这种序列的半经典波前集$WF_h$在Hamilton流下是不变的。在原则上，这可能会允许浓度的$WF_h$沿正余维子环面的刘维环面$\mathcal{L}$如果存在合理的频率之间的关系流$\mathcal{L}.$我们表明，受非简并假设，这种浓度实际上可能不会发生。主要的工具是由Vasy和作者先前所示的$\mathcal{L}$上的拉格朗日正则性的扩展，以及拉格朗日准模的主象征所满足的高阶输运方程的分析。
英文摘要：
We consider the possible concentration in phase space of a sequence of eigenfunctions (or, more generally, a quasimode) of an operator whose principal symbol has completely integrable Hamilton flow. The semiclassical wavefront set $WF_h$ of such a sequence is invariant under the Hamilton flow. In principle this may allow concentration of $WF_h$ along positive codimension sub-tori of a Liouville torus $\mathcal{L}$ if there exist rational relations among the frequencies of the flow on $\mathcal{L}.$ We show that, subject to non-degeneracy hypotheses, this concentration may not in fact occur. The main tools are the spreading of Lagrangian regularity on $\mathcal{L}$ previously shown by Vasy and the author, and an analysis of higher order transport equations satisfied by the principal symbol of a Lagrangian quasimode.
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