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

Masamune, Jun

doi:10.1090/S0002-9939-2011-10965-1

On an inclusion of the essential spectrum of Laplacians under non-compact change of metric
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Proc. Amer. Math. Soc. 140 (2012), 1045-1052 Request permission

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

Shigetoshi Bando and Hajime Urakawa, Generic properties of the eigenvalue of the Laplacian for compact Riemannian manifolds, Tohoku Math. J. (2) 35 (1983), no. 2, 155–172. MR 699924, DOI 10.2748/tmj/1178229047
Yves Colin de Verdière, Pseudo-laplaciens. I, Ann. Inst. Fourier (Grenoble) 32 (1982), no. 3, xiii, 275–286 (French, with English summary). MR 688031
Masatoshi Fukushima, Y\B{o}ichi Ōshima, and Masayoshi Takeda, Dirichlet forms and symmetric Markov processes, De Gruyter Studies in Mathematics, vol. 19, Walter de Gruyter & Co., Berlin, 1994. MR 1303354, DOI 10.1515/9783110889741
Kenrô Furutani, On a stability of essential spectra of Laplace operators on noncompact Riemannian manifolds, Proc. Japan Acad. Ser. A Math. Sci. 56 (1980), no. 9, 425–428. MR 603058
Matthew P. Gaffney, A special Stokes’s theorem for complete Riemannian manifolds, Ann. of Math. (2) 60 (1954), 140–145. MR 62490, DOI 10.2307/1969703
Matthew P. Gaffney, Hilbert space methods in the theory of harmonic integrals, Trans. Amer. Math. Soc. 78 (1955), 426–444. MR 68888, DOI 10.1090/S0002-9947-1955-0068888-1
Jun Masamune, Essential self-adjointness of Laplacians on Riemannian manifolds with fractal boundary, Comm. Partial Differential Equations 24 (1999), no. 3-4, 749–757. MR 1683058, DOI 10.1080/03605309908821442
Jun Masamune and Wayne Rossman, Discrete spectrum and Weyl’s asymptotic formula for incomplete manifolds, Minimal surfaces, geometric analysis and symplectic geometry (Baltimore, MD, 1999) Adv. Stud. Pure Math., vol. 34, Math. Soc. Japan, Tokyo, 2002, pp. 219–229. MR 1925741, DOI 10.2969/aspm/03410219
Jun Masamune, Analysis of the Laplacian of an incomplete manifold with almost polar boundary, Rend. Mat. Appl. (7) 25 (2005), no. 1, 109–126. MR 2142127