Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

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

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Degeneration of pseudo-Laplace operators for hyperbolic Riemann surfaces
HTML articles powered by AMS MathViewer

by Lizhen Ji PDF
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 $l \to 0$.
  • Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957
  • Isaac Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984. Including a chapter by Burton Randol; With an appendix by Jozef Dodziuk. MR 768584
  • Yves Colin de Verdière, Pseudo-laplaciens. I, Ann. Inst. Fourier (Grenoble) 32 (1982), no. 3, xiii, 275–286 (French, with English summary). MR 688031
  • Yves Colin de Verdière, Pseudo-laplaciens. II, Ann. Inst. Fourier (Grenoble) 33 (1983), no. 2, 87–113 (French). MR 699488
  • J.-M. Deshouillers, H. Iwaniec, R. S. Phillips, and P. Sarnak, Maass cusp forms, Proc. Nat. Acad. Sci. U.S.A. 82 (1985), no. 11, 3533–3534. MR 791741, DOI 10.1073/pnas.82.11.3533
  • David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften, Vol. 224, Springer-Verlag, Berlin-New York, 1977. MR 0473443
  • Dennis A. Hejhal, Regular $b$-groups, degenerating Riemann surfaces, and spectral theory, Mem. Amer. Math. Soc. 88 (1990), no. 437, iv+138. MR 1052555, DOI 10.1090/memo/0437
  • Lizhen Ji, Spectral degeneration of hyperbolic Riemann surfaces, J. Differential Geom. 38 (1993), no. 2, 263–313. MR 1237486
  • Peter D. Lax and Ralph S. Phillips, Scattering theory for automorphic functions, Annals of Mathematics Studies, No. 87, Princeton University Press, Princeton, N.J., 1976. MR 0562288
  • R. S. Phillips and P. Sarnak, On cusp forms for co-finite subgroups of $\textrm {PSL}(2,\textbf {R})$, Invent. Math. 80 (1985), no. 2, 339–364. MR 788414, DOI 10.1007/BF01388610
  • Peter Sarnak, On cusp forms, The Selberg trace formula and related topics (Brunswick, Maine, 1984) Contemp. Math., vol. 53, Amer. Math. Soc., Providence, RI, 1986, pp. 393–407. MR 853570, DOI 10.1090/conm/053/853570
  • A. Selberg, Harmonic analysis (Göttingen Lecture notes), Atle Selberg’s Collected Papers, Springer-Verlag, Berlin and New York, 1989, pp. 626-674.
  • Michael Wolf, Infinite energy harmonic maps and degeneration of hyperbolic surfaces in moduli space, J. Differential Geom. 33 (1991), no. 2, 487–539. MR 1094467
  • Scott A. Wolpert, Asymptotics of the spectrum and the Selberg zeta function on the space of Riemann surfaces, Comm. Math. Phys. 112 (1987), no. 2, 283–315. MR 905169
  • —, The spectrum of a Riemann surface with a cusp, Taniguchi Symposium Lecture, November 1989, preprint.
  • Scott A. Wolpert, Spectral limits for hyperbolic surfaces. I, II, Invent. Math. 108 (1992), no. 1, 67–89, 91–129. MR 1156387, DOI 10.1007/BF02100600
  • —, Spectral limits for hyperbolic surfaces. II, Invent. Math. 108 (1992), 91-129.
  • Scott A. Wolpert, Disappearance of cusp forms in special families, Ann. of Math. (2) 139 (1994), no. 2, 239–291. MR 1274093, DOI 10.2307/2946582
Similar Articles
  • Retrieve articles in Proceedings of the American Mathematical Society with MSC: 58G25
  • Retrieve articles in all journals with MSC: 58G25
Additional Information
  • © Copyright 1994 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 121 (1994), 283-293
  • MSC: Primary 58G25
  • DOI:
  • MathSciNet review: 1184082