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On Rellich's Theorem Concerning Infinitely Narrow Tubes

Published online by Cambridge University Press:  20 November 2018

Colin Clark
Colin Clark*
Affiliation:
University of British Columbia
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Let G be a region in Eućlidean n-space En and consider the eigenvalue problem Δ2u = λu on G, with boundary conditions u = 0 on Γ, the boundary of G. (To be precise, we are considering the eigenvalue problem for the self-adjoint 2 realization L associated with the Laplacian -Δ2and zero boundary condition, acting in L2(G), cf Browder [2]). If G is bounded, the spectrum of this problem is discrete, but Rellich showed in 1952 [6] that the spectrum could also be discrete for certain unbounded regions which he introduced and called "infinitely narrow tubes".

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 7 , Issue 3 , September 1964 , pp. 435 - 440
Copyright
Copyright © Canadian Mathematical Society 1964

References

1. Brownell, F. H. and Clark, C. W., Asymptotic distribution of the eigenvalues for the lower part of the Schrődinger operator spectrum, Jour. Math. Mech. 10 (1961), pp. 31–70.Google Scholar
2. Browder, F. E., On the spectral theory of elliptic differential operators 1, Math. An. 142 (1961), pp. 22–130.Google Scholar
3. Courant, and Hilbert, , Methods of Mathematical Physics, Vol.I; Interscience, N. Y. (1953).Google Scholar
4. Jones, D.S., The eigenvalues of Δ2 u + λ u = 0 when the boundary conditions are given on semi-infinite domains, Proc. Cambridge Philos. Soc. 49 (1953), pp. 668–684.Google Scholar
5. Molcanov, A.M., On conditions for discreteness of the spectrum of self-adjoint differential equations of the second order (Russian), Mosk. Mat. Obschestva (Trudy) 2 (1953), pp. 169–199.Google Scholar
6. Rellich, F., Das Eigenwertproblem von Δu + λu=0 in Halbrőhren, in Essays Presented to R. Courant, N. Y. (1948), pp. 329–344.Google Scholar