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An addition theorem for the manifolds with the Laplacian having discrete spectrum. (Russian, English) Zbl 1115.58028
Sib. Mat. Zh. 47, No. 3, 557-574 (2006); translation in Sib. Math. J. 47, No. 3, 459-473 (2006).
Summary: The question of the preservation of discreteness of the spectrum of the Laplacian acting in a space of differential forms under the cutting and gluing of manifolds reduces to the same problem for compact solvability of the operator of the exterior derivation. Along these lines, we give some conditions on a cut \(Y\) dividing a Riemannian manifold \(X\) into two parts \(X_+\) and \(X_-\) under which the spectrum of the Laplacian on \(X\) is discrete if and only if so are the spectra of the Laplacians on \(X_+\) and \(X_-\).
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
35P05 General topics in linear spectral theory for PDEs
47F05 General theory of partial differential operators
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