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On an inclusion of the essential spectrum of Laplacians under non-compact change of metric


Author: Jun Masamune
Journal: Proc. Amer. Math. Soc. 140 (2012), 1045-1052
MSC (2010): Primary 58J50; Secondary 47B25
DOI: https://doi.org/10.1090/S0002-9939-2011-10965-1
Published electronically: June 29, 2011
MathSciNet review: 2869089
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Abstract: The stability of essential self-adjointness and an inclusion of the essential spectra of Laplacians under the change of a Riemannian metric on a subset $ K$ of $ M$ are proved. The set $ K$ may have infinite volume measured with the new metric, and its completion may contain a singular set such as the fractal set, to which the metric is not extendable.


References [Enhancements On Off] (What's this?)

  • 1. Bando, S.; and Urakawa, H., Generic properties of the eigenvalue of the Laplacian for compact Riemannian manifolds. Tohoku Math. J. (2) 35 (1983), no. 2, 155-172. MR 699924 (84h:58146)
  • 2. Colin de Verdière, Yves, Pseudo-laplaciens. I. Ann. Inst. Fourier (Grenoble) 32 (1982), no. 3, xiii, 275-286. MR 688031 (84k:58221)
  • 3. Fukushima, M; Oshima, Y.; Takeda, M., Dirichlet forms and symmetric Markov processes. de Gruyter Studies in Mathematics, 19. Walter de Gruyter & Co., Berlin, 1994. MR 1303354 (96f:60126)
  • 4. Furutani, K., On a stability of essential spectra of Laplace operators on non-compact Riemannian manifolds. Proc. Japan Acad. 56, Ser. A (1980), 425-428. MR 603058 (82c:58055)
  • 5. Gaffney, M.P., A special Stokes's theorem for complete Riemannian manifolds. Ann. of Math. (2) 60 (1954), 140-145. MR 0062490 (15:986d)
  • 6. Gaffney, M.P., Hilbert space methods in the theory of harmonic integrals. Trans. Amer. Math. Soc. 78 (1955), 426-444. MR 0068888 (16:957a)
  • 7. Masamune, J., Essential self-adjointness of Laplacians on Riemannian manifolds with fractal boundary. Comm. Partial Differential Equations 24 (1999), no. 3-4, 749-757. MR 1683058 (2000m:58035)
  • 8. Masamune, J.; Rossman, W., Discrete spectrum and Weyl's asymptotic formula for incomplete manifolds. Minimal surfaces, geometric analysis and symplectic geometry (Baltimore, MD, 1999), 219-229, Adv. Stud. Pure Math., 34, Math. Soc. Japan, Tokyo, 2002. MR 1925741 (2004b:58052)
  • 9. Masamune, J., Analysis of the Laplacian of an incomplete manifold with almost polar boundary. Rend. Mat. Appl. (7) 25 (2005), no. 1, 109-126. MR 2142127 (2006a:58040)

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Additional Information

Jun Masamune
Affiliation: Department of Mathematical Sciences, Worcester Polytechnic Institute, 100 Institute Road, Worcester, Massachusetts 01609-2280
Address at time of publication: Department of Mathematics and Statistics, Pennsylvania State University-Altoona, 3000 Ivyside Park, Altoona, Pennsylvania 16601
Email: jum35@psu.edu

DOI: https://doi.org/10.1090/S0002-9939-2011-10965-1
Keywords: Essential self-adjointness, incomplete manifolds, essential spectrum, perturbations
Received by editor(s): March 30, 2010
Received by editor(s) in revised form: September 23, 2010, and December 8, 2010
Published electronically: June 29, 2011
Communicated by: Matthew J. Gursky
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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