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• W2852785493 abstract "We prove Szegő-type trace asymptotics for translation-invariant operators on polygons. More precisely, consider a Fourier multiplier $A=mathcal{F}^ast sigma mathcal{F}$ on $mathsf{L}^2(mathbb{R}^2)$ with a sufficiently decaying, smooth symbol $sigma:mathbb{C}tomathbb{C}$. Let $Psubset mathbb{R}^2$ be the interior of a polygon and, for $Lgeq 1$, define its scaled version $P_L:=Lcdot P$. Then we study the spectral asymptotics for the operator $A_{P_L}=chi_{P_L}Achi_{P_L}$, 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 $Ltoinfty$. 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." @default.
• W2852785493 created "2018-07-19" @default.
• W2852785493 creator A5076975575 @default.
• W2852785493 date "2018-07-12" @default.
• W2852785493 modified "2023-09-27" @default.
• W2852785493 title "A SzegH{o} limit theorem for translation-invariant operators on polygons" @default.
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• W2852785493 hasPublicationYear "2018" @default.
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