Abstract
We investigate the vector bundle of the manifold of normalized eigenvectors of self-adjoint operators and its stratification with respect to the numbers and multiplicities of eigenvalues.
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REFERENCES
I. V. Skrypnik, Methods for the Investigation of Nonlinear Elliptic Boundary-Value Problems [in Russian], Nauka, Moscow (1990).
L. A. Lyusternik, “On one boundary-value problem in the theory of nonlinear differential equations,” Dokl. Akad. Nauk SSSR, 33, 5–8 (1941).
Ya. M. Dymarskii, “Existence, oscillation properties, and asymptotics of normalized eigenfunctions of nonlinear boundary-value problems,” in: Qualitative and Approximate Methods for the Investigation of Operator Equations [in Russian], Yaroslavl (1985), pp. 133–139.
C. Cosner, “Bifurcations from higher eigenvalues,” Nonlin. Anal., 12, No. 3, 271–277 (1988).
Yu. G. Borisovich and O. V. Kunakovskaya, “Intersection theory methods,” in: Abstracts of the Conference on Stochastic and Global Analysis, Voronezh (1996), pp. 10–12.
I. M. Gel'fand, Lectures on Nonlinear Algebra [in Russian], Nauka, Moscow (1971).
Ya. M. Dymarskii, “On the Lyusternik theorem for a two-point problem of the second order,” in: Qualitative and Approximate Methods for the Investigation of Operator Equations [in Russian], Yaroslavl (1984), pp. 16–24.
Ya. M. Dymarskii, “On typical bifurcations in one class of operator equations,” Dokl. Ros. Akad. Nauk, 338, No. 4, 446–449 (1994).
Ya. M. Dymarskii, “On branches of small solutions of certain operator equations,” Ukr. Mat. Zh., 48, No. 7, 901–909 (1996).
Ya. M. Dymarskii, “On normalized eigenfunctions of one class of quasilinear elliptic equations,” Differents. Uravn., 34, No. 1, 127–129 (1998).
V. I. Arnol'd, Mathematical Methods of Classical Mechanics [in Russian], Nauka, Moscow (1974).
V. I. Arnol'd, “Remarks on the eigenvalues and eigenvectors of Hermitian matrices,” in: V. I. Arnol'd, Collected Works [in Russian], Fazis, Moscow (1998), pp. 583–604.
B. A. Dubrovin, S. P. Novikov, and A. T. Fomenko, Modern Geometry [in Russian], Nauka, Moscow (1979).
A. T. Fomenko and D. B. Fuks, A Course of Homotopic Topology [in Russian], Nauka, Moscow (1989).
K. Uhlenbeck, “Generic properties of eigenfunctions,” Amer. J. Math., 98, No. 4, 1059–1078 (1976).
D. Husemoller, Fibre Bundles [Russian translation], Mir, Moscow (1970).
B. A. Dubrovin, S. P. Novikov, and A. T. Fomenko, Modern Geometry. Methods of the Theory of Homologies [in Russian], Nauka, Moscow (1984).
V. A. Rokhlin and D. B. Fuks, Beginner's Course in Topology [in Russian], Nauka, Moscow (1977).
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Dymarskii, Y.M. On the Manifolds of Eigenvectors of Linear and Quasilinear Finite-Dimensional Self-Adjoint Operators. I. Ukrainian Mathematical Journal 53, 178–189 (2001). https://doi.org/10.1023/A:1010404717849
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DOI: https://doi.org/10.1023/A:1010404717849