2018/12/30 by A. Yu. Savin, Anton Savin, Elmar Schrohe
Mathematics · #Abelian group #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Algebra over a field #Algebraic cycle #Algebraic number #Corollary #Elliptic operator #Fourier integral operator #Mathematical analysis #Mathematics #Operator (biology) #Operator theory #Pure mathematics #Quantum #Semiclassical physics #Spectral Theory in Mathematical Physics #TRACE (psycholinguistics) #math.AP #math.OA #msc:46L87 #msc:58J20 #msc:58J40
paper · pdf · doi:10.1016/j.jfa.2019.108400
published as Journal of Functional Analysis, Volume 278, Issue 5, 2020, 108400
arxiv created 2018/12/30 · openalex publication_date 2019/11/13 · arxiv updated 2020/08/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We consider elliptic operators associated with discrete groups of quantized canonical transformations. In order to be able to apply results from algebraic index theory, we define the localized algebraic index of the complete symbol of an elliptic operator. With the help of a calculus of semiclassical quantized canonical transformations, a version of Egorov's theorem and a theorem on trace asymptotics for semiclassical Fourier integral operators we show that the localized analytic index and the localized algebraic index coincide. As a corollary, we express the Fredholm index in terms of the algebraic index for a wide class of groups, in particular, for finite extensions of Abelian groups.