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Eigenvalues of the Laplacian for rectilinear regions

Published online by Cambridge University Press:  17 February 2009

H. P. W. Gottlieb
H. P. W. Gottlieb
Affiliation:
School of Science, Griffith University, Nathan, Queensland 4111, Australia.
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Abstract

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From a knowledge of the eigenvalue spectrum of the Laplacian on a domain, one may extract information on the geometry and boundary conditions by analysing the asymptotic expansion of a spectral function. Explicit calculations are performed for isosceles right-angle triangles with Dirichlet or Neumann boundary conditions, yielding in particular the corner angle terms. In three dimensions, right prisms are dealt with, including the solid vertex terms.

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 29 , Issue 3 , January 1988 , pp. 270 - 281
Copyright
Copyright © Australian Mathematical Society 1988

References

[1]Baltes, H. P., “Asymptotic eigenvalue distribution for the wave equation in a cylinder of arbitrary cross section”, Phys. Rev. A 6 (1972) 2252–2257.CrossRefGoogle Scholar
[2]Brossard, J. and Carmona, R., “Can one hear the dimension of a fractal?”, Comm. Math. Phys. 104 (1986) 103–122.CrossRefGoogle Scholar
[3]Gottlieb, H. P. W., “Hearing the shape of an annular drum”, J. Austral. Math. Soc. Ser. B 24 (1983) 435–438.CrossRefGoogle Scholar
[4]Gottlieb, H. P. W., “Eigenvalues of the Laplacian with Neumann boundary conditions”, J. Austral. Math. Soc. Ser. B 26 (1985) 293–309.CrossRefGoogle Scholar
[5]Hansen, E. R., A table of series and products (Prentice-Hall, Englewood Cliffs, 1975).Google Scholar
[6]Kac, M., “Can one hear the shape of a drum?”, Amer. Math. Monthly 73 (1966) 1–23.CrossRefGoogle Scholar
[7]McKean, H. P. and Singer, I. M., “Curvature and the eigenvalues of the Laplacian”, J. Differential Geom. 1 (1967) 43–69.CrossRefGoogle Scholar
[8]Morse, P. M. and Feshbach, H., Methods of theoretical physics, Vol. 1 (McGraw-Hill, New York, 1953).Google Scholar
[9]Pleijel, A., “A study of certain Green's functions with applications in the theory of vibrating membranes”, Ark. Mat. 2 (1953–1954) 553–569.CrossRefGoogle Scholar
[10]Sleeman, B. D., “The inverse problem of acoustic scattering”, IMA J. Appl. Math. 29 (1982) 113–142.CrossRefGoogle Scholar
[11]Sleeman, B. D. and Zayed, E. M. E., “An inverse eigenvalue problem for a general convex domain”, J. Math. Anal. Appl. 94 (1983) 78–95.CrossRefGoogle Scholar
[12]Sleeman, B. D. and Zayed, E. M. E., “Trace formula for the eigenvalues of the Laplacian”, ZAMP J. Appl. Math. Phys. 35 (1984) 106–115.Google Scholar
[13]Stewartson, K. and Waechter, R. T., “On hearing the shape of a drum: further results”, Proc. Cambridge Philos. Soc. 69 (1971) 353–363.CrossRefGoogle Scholar
[14]Urakawa, H., “Bounded domains which are isospectral but not congruent”, Ann. Sci. École Norm. Sup. 15 (1982) 441–456.CrossRefGoogle Scholar
[15]Waechter, R. T., “On hearing the shape of a drum: an extension to higher dimensions”, Proc. Cambridge Philos. Soc. 72 (1972) 439–447.CrossRefGoogle Scholar
[16]Zayed, E. M. E., “An inverse eigenvalue problem for the Laplace operator”, in Ordinary and partial differential equations, (eds. Everitt, W. N. and Sleeman, B. D.), (Springer, Berlin, 1982) 718–726.CrossRefGoogle Scholar
[17]Zayed, E. M. E., “Eigenvalues of the Laplacian: an extension to higher dimensions”, IMA J. Appl. Math. 33 (1984) 83–99.CrossRefGoogle Scholar