Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-10T20:58:48.246Z Has data issue: false hasContentIssue false
  • English
  • Français

Spectral Theory for the Neumann Laplacian on Planar Domains With Horn-Like Ends

Published online by Cambridge University Press:  20 November 2018

Julian Edward
Julian Edward*
Affiliation:
Department of Mathematics, Florida International University, Miami, Florida 33199, U.S.A. e-mail: edwardj@fiu.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The spectral theory for the Neumann Laplacian on planar domains with symmetric, horn-like ends is studied. For a large class of such domains, it is proven that the Neumann Laplacian has no singular continuous spectrum, and that the pure point spectrum consists of eigenvalues of finite multiplicity which can accumulate only at 0 or ∞. The proof uses Mourre theory.

Keywords

35P2558G25
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 49 , Issue 2 , 01 April 1997 , pp. 232 - 262
Copyright
Copyright © Canadian Mathematical Society 1997

References

1. Berger, G., Asymptotische Eigenwertverteilung des Laplace-Operators in bestimmten unbeschrankten Gebieten mit Neumannschen Randbedingungen und Restgliedabschatzungen, Z. Anal. Anwendungen 4(1985), 85–96.Google Scholar
2. Cycon, H.L., Froese, R.G., Kirsch, W., and Simon, B., Schrodinger Operators, with Applications to Spectral Geometry, Springer Texts and Monographs in Physics, Springer-Verlag, Berlin, 1987.Google Scholar
3. Davies, E.B. and Simon, B., Spectral properties of the Neumann Laplacian on horns, Geom. Funct. Anal. (1) 2(1992), 105–117.Google Scholar
4. Edward, J., Spectral theory of the Neumann Laplacian on asymptotically perturbed waveguides, submitted.Google Scholar
5. Evans, W.D. and Harris, D.J., Sobolev embeddings for generalised ridge domains, Proc. London Math. Soc. (3) 54(1987), 141–175.Google Scholar
6. Froese, R.G. and Hislop, P., Spectral analysis of second order elliptic operators on non-compact manifolds, Duke Math. J. 58(1989), 103–129.Google Scholar
7. Hempel, R., Seco, L., and Simon, B., The essential spectrum of Neumann Laplacians on some bounded singular domains, J. Funct. Anal. 102(1991), 448–483.Google Scholar
8. Jaksic, V., On the Spectrum of Neumann Laplacian of Long Range Horns: A note on Davies-Simon Theorem, Proc. Amer.Math. Soc. 119(1993), 663–669.Google Scholar
9. Jaksic, V., Molcanov, S., and Simon, B., Eigenvalue asymptotics of the Neumann Laplacian of regions and manifolds with cusps, J. Funct. Anal. (1) 106(1992), 59–79.Google Scholar
10. Mourre, E., Absence of singular continuous spectrum for certain self-adjoint operators, Comm. Math. Phys. 78(1981), 391–408.Google Scholar
11. Reed, M. and Simon, B., Methods of Modern Mathematical Physics, Vols. 1–4. New York, Academic Press, 1972.Google Scholar