多边形上的热迹异常
A heat trace anomaly on polygons
摘要：
令$Ω_0$是$\RR^2$中的多边形，或者更一般地是具有分段光滑边界和角的紧凑曲面。假设$Ω_\e$是一个曲面族，其$\calC^\infty$边界在远离角点处光滑地收敛到$Ω_0$，并且在本文所描述的顶点处以精确的方式收敛。Fedosov \cite{Fe}、Kac \cite{K}和McKean-Singer \cite{MS}认识到某些热迹系数，特别是$t^0$的系数，作为$\e \searrow 0$是不连续的。我们描述这种异常使用重整化热不变量的辅助光滑域$Z$模型的角落形成。结果适用于狄利克雷和诺依曼条件。我们还包括一个讨论，人们可能会期望在更高的维度。
英文摘要：
Let $Ω_0$ be a polygon in $\RR^2$, or more generally a compact surface with piecewise smooth boundary and corners. Suppose that $Ω_\e$ is a family of surfaces with $\calC^\infty$ boundary which converges to $Ω_0$ smoothly away from the corners, and in a precise way at the vertices to be described in the paper. Fedosov \cite{Fe}, Kac \cite{K} and McKean-Singer \cite{MS} recognized that certain heat trace coefficients, in particular the coefficient of $t^0$, are not continuous as $\e \searrow 0$. We describe this anomaly using renormalized heat invariants of an auxiliary smooth domain $Z$ which models the corner formation. The result applies both for Dirichlet and Neumann conditions. We also include a discussion of what one might expect in higher dimensions.
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