Premium
A Szegő limit theorem for translation‐invariant operators on polygons
Author(s) -
Pfirsch Bernhard
Publication year - 2019
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201800325
Subject(s) - mathematics , polygon (computer graphics) , laplace operator , asymptotic expansion , invariant (physics) , trace (psycholinguistics) , mathematical analysis , dirichlet distribution , differential operator , multiplier (economics) , boundary value problem , mathematical physics , telecommunications , linguistics , philosophy , frame (networking) , computer science , economics , macroeconomics
We prove Szegő‐type trace asymptotics for translation‐invariant operators on polygons. More precisely, consider a Fourier multiplier A = F * σ F onL 2 ( R 2 )with a sufficiently decaying, smooth symbol σ : R 2 → C . Let P ⊂ R 2be the interior of a polygon and, for L ≥ 1 , define its scaled versionP L : = L · P . Then we study the spectral asymptotics for the operatorA P L = χ P L A χ 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 tr h (A P L)as L → ∞ . 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.