Trace-formula methods in spectral geometry
type: Item
instance of: scholarly article
title: Trace-formula methods in spectral geometry
zbMATH Open document ID: 0894.58073
author: Robert Brooks
published in: The Journal of Fourier Analysis and Applications
publication date: 1998-09-07T00:00:00Z
review text: Let \(\Gamma\) be the modular group acting on the upper half-plane \(\mathbb{H}\) and denote by \(\Gamma_n\) the principal congruence subgroup of \(\Gamma\). If \(\lambda_1\) denotes the least positive eigenvalue of \(-\Delta\) on \(\Gamma_n \setminus \mathbb{H}\), then a famous theorem of Selberg asserts that \(\lambda_1 \geq {3\over 16}\) and a famous conjecture of Selberg claims that \(\lambda_1 \geq {1\over 4}\). The paper under review arose from the question of understanding Selberg's theorem in a geometric way. First, the author proves a trace formula for graphs. An application of this trace formula yields an analogue of the Selberg \({3 \over 16}\)-theorem for graphs. This theorem is then adapted to Riemann surfaces and yields an asymptotic bound on \(\lambda_1\) for certain families of Riemann surfaces which have many properties in common with the family of congruence surfaces.
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 11F72
Mathematics Subject Classification ID: 05C35
Mathematics Subject Classification ID: 05C99
zbMATH DE Number: 1115808
zbMATH Keywords: trace formula
zbMATH Keywords: eigenvalue
zbMATH Keywords: graph
zbMATH Keywords: Riemann surface
zbMATH Keywords: Selberg conjecture
reviewed by: Jürgen Elstrodt
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1115808