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 $(> \frac {1}{4})$ 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 ${S_l} (l \geq 0)$, we show that normalized pseudoeigenfunctions and pseudoeigenvalues of ${S_l}$ converge to normalized pseudoeigenfunctions and pseudoeigenvalues of ${S_0}$ as $l \to 0$. In particular, normalized embedded eigenfunctions and their embedded eigenvalues of ${S_0}$ can be approximated by normalized pseudoeigenfunctions and pseudoeigenvalues of ${S_l}$ and $l \to 0$.