Asymptotic properties of eigenfunctions -- the hyperbolic plane
Let \(M\) be a complete, noncompact Riemannian manifold. The associated Laplacian \(\Delta\) is an unbounded selfadjoint, nonnegative operator on \(L^ 2 M\). Suppose that the spectrum of \(\Delta\) has a positive lower- bound. If \(M'\) is obtained from \(M\) by a compactly supported perturbation of the metric and topology then it is well known that the Laplacian \(\Delta\) and \(\Delta'\) have the same spectrum, see for details the author, Topology 20, 1-14 (1981; Zbl 0463.53027). However, such perturbations often introduce a finite number of discrete eigenvalues \(\lambda'\), lying below the essential spectrum, with associated eigenfunctions \(\phi'\). The eigenfunctions \(\phi'\) are expected to be especially well behaved, compared to arbitrary eigenfunctions on noncompact manifolds. In the paper under review the author investigates this expectation in the particular case where \(M = H\), the Poincaré plane. The results are similar with those obtained by \textit{M. Hoffmann- Ostenhof, T. Hoffmann-Ostenhof} and \textit{J. Swetina}, Duke Math. J. 53, 271-306 (1986; Zbl 0599.35036) for the Schrödinger operator in the Euclidean plane.
scientific article; zbMATH DE number 640335