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Proceedings of the American Mathematical Society

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ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Degeneration of pseudo-Laplace operators for hyperbolic Riemann surfaces
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Proc. Amer. Math. Soc. 121 (1994), 283-293 Request permission


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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Additional Information
  • © Copyright 1994 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 121 (1994), 283-293
  • MSC: Primary 58G25
  • DOI:
  • MathSciNet review: 1184082