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Degeneration of pseudo-Laplace operators for hyperbolic Riemann surfaces


Author: Lizhen Ji
Journal: Proc. Amer. Math. Soc. 121 (1994), 283-293
MSC: Primary 58G25
DOI: https://doi.org/10.1090/S0002-9939-1994-1184082-5
MathSciNet review: 1184082
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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$.


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DOI: https://doi.org/10.1090/S0002-9939-1994-1184082-5
Article copyright: © Copyright 1994 American Mathematical Society

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