Hadamard's problem and Coxeter groups: New examples of Huygens' equations
type: Item
instance of: scholarly article
title: Hadamard's problem and Coxeter groups: New examples of Huygens' equations
zbMATH Open document ID: 0845.35062
DOI: 10.1007/BF01079005
publication date: 1996-09-11T00:00:00Z
review text: The main result of the work is the following. By a root system we mean a finite set \({\mathfrak R}= \{\alpha\}\subseteq \mathbb{R}^n\) of vectors that have pairwise different directions, span \(\mathbb{R}^n\), and satisfy the condition that the reflections \(s_\alpha(\beta)= \beta- 2\alpha(\alpha, \beta)/(\alpha, \alpha)\) transform \(\mathfrak R\) into itself. The reflections \(s_\alpha\) generate a finite group \(\mathcal W\) referred to as the Coxeter group. Let us define the potential associated with \(\mathcal R\) by setting \[ u(x)= \sum_{\alpha\in {\mathfrak R}_+} {m_\alpha(m_\alpha+ 1)(\alpha, \alpha)\over (\alpha, x)^2}, \] where \(m_\alpha\) are integers such that the function \(m(\alpha)= m_\alpha\) is \({\mathcal W}\)-invariant, and \({\mathfrak R}_+\subset {\mathfrak R}\) denotes the set of roots positive with respect to a suitable linear form on \(\mathbb{R}^n\). Consider the hyperbolic operator \[ {\mathcal L}= {\partial^2\over \partial t^2}- {\partial^2\over \partial x^2_1}-\cdots- {\partial^2\over \partial x^2_n}- {\partial^2\over \partial y^2_1}-\cdots- {\partial^2\over \partial y^2_m}+ u(x_1,\dots, x_n), \] where \(N= m+ n\) is an arbitrary odd number satisfying the inequality \(N\geq 3+ 2 \sum_{\alpha\in {\mathfrak R}_+} m_\alpha\). Theorem. For any Coxeter group \(\mathcal W\) and integer-valued \({\mathcal W}\)-invariant function \(m\) on the corresponding root system \(\mathfrak R\) the equation \({\mathcal L}\varphi= 0\) satisfies Huygens' principle.
Mathematics Subject Classification ID: 35L10
zbMATH DE Number: 859663
zbMATH Keywords: Hadamard's problem
zbMATH Keywords: Coxeter group
zbMATH Keywords: Huygens' principle
author: Yuri Berest
author: A. P. Veselov
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01079005
OpenAlex ID: W2039633458
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 859663