The spectral function of a Riemannian orbifold
Let \(X\) be a compact orbifold and \(P\) a first-order symmetric elliptic pseudodifferential operator on \(X\) with positive principal symbol. It is known that \(P\) has a discrete spectrum and there exists an orthonormal basis of eigenfunctions \(\{\psi_{j}\}\): \(P\psi_{j}=\mu_{j}\psi_{j}\), where the eigenvalues \(\mu_{j}\) are ordered as \(\mu_{1}\leq\mu_{2}\leq\cdots\). In the paper under review, the authors study the asymptotics of the spectral function \(\sum_{j:\mu_{j}\leq\mu}|\psi_{j}(x_{0})|^{2}\) as \(\mu\) goes to infinity, for a fixed \(x_{0}\in X\). It is noted that the formula depends on \(x_{0}\), in contrast with the case where \(X\) is a (Riemannian) manifold.
scientific article; zbMATH DE number 5903717
scientific article