Geodesic
On Rellich's Theorem Concerning Infinitely Narrow Tubes
Canadian mathematical bulletin, Tome 7 (1964) no. 3, pp. 435-440

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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". 
Clark, Colin. On Rellich's Theorem Concerning Infinitely Narrow Tubes. Canadian mathematical bulletin, Tome 7 (1964) no. 3, pp. 435-440. doi: 10.4153/CMB-1964-043-6
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     author = {Clark, Colin},
     title = {On {Rellich's} {Theorem} {Concerning} {Infinitely} {Narrow} {Tubes}},
     journal = {Canadian mathematical bulletin},
     pages = {435--440},
     year = {1964},
     volume = {7},
     number = {3},
     doi = {10.4153/CMB-1964-043-6},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1964-043-6/}
}
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