Abstract
In the paper, we study manifolds of eigenfunctions of a fixed oscillation. Then, solving the trivial inverse problem of reconstruction of a potential by an eigenfunction, we describe the properties of manifolds of potentials. The approach proposed allows one to link topological properties of manifolds of eigenfunctions with those of manifolds of potentials.
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References
V. I. Arnold, “Modes and quasimodes,” Funkts. Anal. Prilozh., 6, No. 2, 94–101 (1972).
V. I. Arnold, Selected Works [in Russian], Fazis Press, Moscow (1998).
J.-P. Bourguignon, “The Sturm-Liouville equation whose solutions are all periodic,” in: A. Besse, Manifolds All of Whose Geodesics Are Closed, Springer-Verlag, Berlin (1978), pp. 290–305.
Ya. M. Dymarskii, “On manifolds of eigenfunctions and potentials generated by a family of periodic boundary-value problems,” Ukr. Math. J., 48, No. 6, 866–879 (1996).
Ya. M. Dymarskii, “On manifolds of self-adjoint elliptic operators with multiple eigenvalues,” Methods Funct. Anal. Topology, 7, No. 2, 68–74 (2001).
Ya. M. Dymarskii, “On manifolds of eigenfunctions and potentials of the family of periodic Sturm-Liouville problems,” Ukr. Mat. Zh., 54, No. 8, 1042–1052 (2002); English transl. in Ukr. Math. J., 54, No. 8, 1251–1263 (2002).
L. D. Landau and E. M. Lifshitz, Theoretical Physics. Vol. 3: Quantum Mechanics. Nonrelativistic Theory, Pergamon Press, Oxford (1977).
F. Neuman, “Linear differential equations of the second order and their application,” Rend. Mat. (6), 4, 559–617 (1971).
M. Reed and B. Simon, Methods of Modern Mathematical Physics IV: Analysis of Operators, Academic Press, New York (1978).
K. Uhlenbeck, “Generic properties of eigenfunctions,” Amer. J. Math., 98, No. 4, 1059–1078 (1976).
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Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 16, Differential and Functional Differential Equations. Part 2, 2006.
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Dymarskii, Y.M. On topological properties of manifolds of eigenfunctions generated by a family of periodic Sturm-Liouville problems. J Math Sci 149, 1488–1503 (2008). https://doi.org/10.1007/s10958-008-0078-0
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DOI: https://doi.org/10.1007/s10958-008-0078-0