Degeneration of pseudo-Laplace operators for hyperbolic Riemann surfaces

Abstract

For finite volume, noncompact Riemann surfaces with their canonical hyperbolic metrics, there is a notion of pseudo-Laplace operators which include all embedded eigenvalues ( > 1 4 ) (> \frac {1}{4}) of the Laplacian as a part of their eigenvalues. Similarly, we define pseudo-Laplace operators for compact hyperbolic Riemann surfaces with short geodesics. Then, for any degenerating family of hyperbolic Riemann surfaces S l ( l ≥ 0 ) {S_l} (l \geq 0) , we show that normalized pseudoeigenfunctions and pseudoeigenvalues of S l {S_l} converge to normalized pseudoeigenfunctions and pseudoeigenvalues of S 0 {S_0} as l → 0 l \to 0 . In particular, normalized embedded eigenfunctions and their embedded eigenvalues of S 0 {S_0} can be approximated by normalized pseudoeigenfunctions and pseudoeigenvalues of S l {S_l} and l → 0 l \to 0 .