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 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 −.
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Glazman I. M., Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators [in Russian], Fizmatgiz, Moscow (1963).
Kuz’minov V. I. and Shvedov I. A., “On compact solvability of the operator of exterior derivation,” Siberian Math. J., 38, No. 3, 492–506 (1997).
Kato T., Perturbation Theory for Linear Operators [Russian translation], Mir, Moscow (1972).
Kuz’minov V. I. and Shvedov I. A., “Homological aspects of the theory of Banach complexes,” Siberian Math. J., 40, No. 4, 754–763 (1999).
Gol’dshtein V. M., Kuz’minov V. I., and Shvedov I. A., “A property of de Rham regularization operators,” Siberian Math. J., 25, No. 2, 251–257 (1984).
Gaffney M. P., “The harmonic operator for exterior differential forms,” Proc. Nat. Acad. Sci., 37, 48–50 (1951).
Schwartz L., Complex Analytic Manifolds. Elliptic Partial Differential Equations [Russian translation], Mir, Moscow (1964).
Gol’dshtein V. M., Kuz’minov V. I., and Shvedov I. A., “Integral representation of the integral of a differential form,” in: Functional Analysis and Mathematical Physics [in Russian], Inst. Mat. (Novosibirsk), Novosibirsk, 1985, pp. 53–87.
Gol’dshtein V. M., Kuz’minov V. I., and Shvedov I. A., “On normal and compact solvability of linear operators,” Siberian Math. J., 30, No. 5, 704–712 (1989).
Gol’dshtein V. M., Kuz’minov V. I., and Shvedov I. A., “The L p-cohomology of Riemannian manifolds,” in: Studies in Geometry and Mathematical Analysis [in Russian], Trudy Inst. Mat. Vol. 7 (Novosibirsk), Nauka, Novosibirsk, 1987, pp. 101–116.
Eichhorn J., “Spektraltheorie offener Riemannscher Mannigfaltigkeiten mit einer rotationssymmetrischen Metrik,” Math. Nachr., Bd 144, 23–51 (1983).
Original Russian Text Copyright © 2006 Kuz’minov V. I. and Shvedov I. A.
The authors were supported by the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grant NSh-311.2003.1).
Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 47, No. 3, pp. 557–574, May–June, 2006.
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Kuz’minov, V.I., Shvedov, I.A. An addition theorem for the manifolds with the Laplacian having discrete spectrum. Sib Math J 47, 459–473 (2006). https://doi.org/10.1007/s11202-006-0058-x
- differential form