Asymptotics of the discrete spectrum of the Dirichlet and Neumann problems on the fundamental domain of the crystallographic group (Q916906)

Let \(F\subset {\mathbb{R}}^ n\) be a regular polyhedron (i.e. all angles between edges of f are quotients of \(\pi\)). Then F is a fundamental domain of group \(\Gamma\), generated by reflections from all edges of F. It allows us to apply the Levitan's wave equation method in order to calculate the asymptotic behaviour of the discrete spectrum of Dirichlet and Neumann problems in F. We obtain two terms of the asymptotic of \(N(\lambda):=\sum_{\lambda_ i\leq \lambda}1\) \((\lambda_ i\) are eigenvalues): \[ N(\lambda^ 2)=C_ n| F| \lambda^ n\pm C'_ n| \partial F| \lambda^{n-1}+o(\lambda^{n-1}), \] where \(``+''\) and ``-'' should be taken in case of Neumann and Dirichlet problems, respectively. We also obtain n terms of the average of N(\(\lambda\)) (i.e. \(\int^{\lambda}_{-\lambda}(1+\frac{t}{\lambda})^ sdN(t)\), where s is sufficiently large); the coefficient near the k-th term is generalized perimeter F of codimension k, i.e. the sum of volumes of all edges of F of codimension k, each taken with some weight dependent only on the angle by this edge.