多边形上平移不变算子的极限定理
A Szegő limit theorem for translation-invariant operators on polygons
摘要：
证明了多边形上平移不变算子的SzegIgn型迹渐近性。更精确地，考虑具有充分衰减的平滑符号$σ:\mathbb{C}\to\mathbb{C}$的$\mathsf{L}^2(\mathbb{R}^2)$上的傅立叶乘法器$A=\mathcal{F}^\ast σ\mathcal{F}$。设$P\subset \mathbb{R}^2$为多边形的内部，对于$L\geq 1$，定义其缩放版本$P_L:=L\cdot P$。然后我们研究了算子$A_{P_L}=χ_{P_L}Aχ_{P_L}$的谱渐近性，$A$对$P_L$的空间限制：对于具有$h(0)=0$的整函数$h$，我们给出了$\operatorname{tr}h(A_{P_L})$作为$L\to\infty$的完全渐近展开式。这些迹渐近由反映多边形几何形状的三项组成。如果$P$被一个光滑边界的域所取代，则30多年来人们已经知道了迹的完全渐近展开。然而，对于多边形的渐近常数阶项的公式是新的。特别是，我们表明，多边形的每个角落产生额外的贡献，因此，恒定的顺序项表现出类似于热跟踪渐近的Dirichlet拉普拉斯算子的异常。
英文摘要：
We prove Szegő-type trace asymptotics for translation-invariant operators on polygons. More precisely, consider a Fourier multiplier $A=\mathcal{F}^\ast σ\mathcal{F}$ on $\mathsf{L}^2(\mathbb{R}^2)$ with a sufficiently decaying, smooth symbol $σ:\mathbb{C}\to\mathbb{C}$. Let $P\subset \mathbb{R}^2$ be the interior of a polygon and, for $L\geq 1$, define its scaled version $P_L:=L\cdot P$. Then we study the spectral asymptotics for the operator $A_{P_L}=χ_{P_L}Aχ_{P_L}$, the spatial restriction of $A$ onto $P_L$: for entire functions $h$ with $h(0)=0$ we provide a complete asymptotic expansion of $\operatorname{tr}h(A_{P_L})$ as $L\to\infty$. These trace asymptotics consist of three terms that reflect the geometry of the polygon. If $P$ is replaced by a domain with smooth boundary, a complete asymptotic expansion of the trace has been known for more than 30 years. However, for polygons the formula for the constant order term in the asymptotics is new. In particular, we show that each corner of the polygon produces an extra contribution; as a consequence, the constant order term exhibits an anomaly similar to the heat trace asymptotics for the Dirichlet Laplacian.
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