Hadamard's problem and Coxeter groups: New examples of Huygens' equations
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.
scientific article; zbMATH DE number 859663
scientific article