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An application of a special form of a Tauberian theorem. (English) Zbl 07125682
Anni, Samuele (ed.) et al., Automorphic forms and related topics. Building bridges: 3rd EU/US summer school and workshop on automorphic forms and related topics, Sarajevo, Bosnia and Herzegovina, July 11–22, 2016. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-3525-7/pbk; 978-1-4704-5317-6/ebook). Contemporary Mathematics 732, 187-193 (2019).
Summary: We are presenting an overview of the applications of a special form of a theorem of Tauberian type to derive new bounds for the remainder term in the Weyl law for the counting function of positive eigenvalues of the Laplacian in three different settings: in the setting of the non-compact, cofinite surfaces of dimension $$d=3$$, in the setting of symmetric spaces of real rank one and in the setting of quantum graphs with general self adjoint boundary conditions.
For the entire collection see [Zbl 1420.11005].
##### MSC:
 11M45 Tauberian theorems 11F72 Spectral theory; trace formulas (e.g., that of Selberg) 34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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