×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bolte, Jens; Endres, Sebastian, The trace formula for quantum graphs with general self adjoint boundary conditions, Ann. Henri Poincar\'e, 10, 1, 189-223 (2009) · Zbl 1207.81028
[2] Friedman, Joshua S., The Selberg trace formula and Selberg zeta-function for cofinite Kleinian groups with finite-dimensional unitary representations, Math. Z., 250, 4, 939-965 (2005) · Zbl 1135.11026
[3] Gangolli, Ramesh; Warner, Garth, Zeta functions of Selberg’s type for some noncompact quotients of symmetric spaces of rank one, Nagoya Math. J., 78, 1-44 (1980)
[4] Hardy, G. H.; Littlewood, J. E., Tauberian Theorems Concerning Power Series and Dirichlet’s Series whose Coefficients are Positive, Proc. London Math. Soc. (2), 13, 174-191 (1914) · JFM 45.0389.02
[5] Korevaar, J., A century of complex Tauberian theory, Bull. Amer. Math. Soc. (N.S.), 39, 4, 475-531 (2002) · Zbl 1001.40007
[6] Od\vzak, Almasa; S\'ceta, Lamija, On the Weyl Law for Quantum Graphs, Bull. Malays. Math. Sci. Soc., 42, 1, 119-131 (2019) · Zbl 1432.11131
[7] Sarnak, P., The arithmetic and geometry of some hyperbolic three-manifolds, Acta Math., 151, 3-4, 253-295 (1983) · Zbl 0527.10022
[8] Smajlovi\'c, Lejla; S\'ceta, Lamija, On a Tauberian theorem with the remainder term and its application to the Weyl law, J. Math. Anal. Appl., 401, 1, 317-335 (2013) · Zbl 1269.40006
[9] Smajlovi\'c, Lejla; S\'ceta, Lamija, On the remainder term in the Weyl law for cofinite Kleinian groups with finite dimensional unitary representation, Arch. Math. (Basel), 102, 2, 117-126 (2014) · Zbl 1376.11045
[10] Tauber, A., Ein Satz aus der Theorie der unendlichen Reihen, Monatsh. Math. Phys., 8, 1, 273-277 (1897) · JFM 28.0221.02
[11] Gangolli, Ramesh; Warner, Garth, Zeta functions of Selberg’s type for some noncompact quotients of symmetric spaces of rank one, Nagoya Math. J., 78, 1-44 (1980)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.