AI 中文总结
本文研究紧致Kähler流形上Bohr–Sommerfeld拉格朗日态的渐近展开，显式计算第一非平凡修正项，并给出$L^2$范数的二阶渐近公式。
AI 中文摘要
设$\Lambda$是紧致Kähler流形中配备全纯预量子线丛的紧致Bohr–Sommerfeld拉格朗日子流形。我们研究与$\Lambda$相关的拉格朗日态的渐近展开。特别地，我们显式地计算了第一个非平凡修正项，并证明它由环境Kähler流形和拉格朗日子流形的几何不变量表示，包括它们的数量曲率、第二基本形式和平均曲率。作为推论，我们获得了拉格朗日态的$L^2$范数的相应二阶渐近公式。
英文摘要
Let $Λ$ be a compact Bohr--Sommerfeld Lagrangian submanifold of a compact Kähler manifold equipped with a holomorphic prequantum line bundle. We study the asymptotic expansion of the Lagrangian states associated with $Λ$. In particular, we compute explicitly the first nontrivial correction term and show that it is expressed in terms of geometric invariants of the ambient Kähler manifold and the Lagrangian submanifold, including their scalar curvatures, the second fundamental form, and the mean curvature. As a consequence, we obtain the corresponding second-order asymptotic formula for the $L^2$-norm of the Lagrangian states.
Comments17 pages
