Abstract
The transverse vibrations of a string with additional restrictions in the form of elastic point constraints are studied. In contrast toWeinstein’s original approach, the constraints are not represented as orthogonality-type conditions in the case under study. Nevertheless, it is shown that the main results of Weinstein’s theory remain valid. It is also shown that the string rigidity coefficients can uniquely be reconstructed from the first-order regularized traces of the corresponding operators. This permits one to give a physical interpretation of regularized traces.
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References
Gould, S., Variational Methods for Eigenvalue Problems: An Introduction to the Weinstein Method of Intermediate Problems, London: Oxford Univ., 1966.
Savchuk, A.M. and Shkalikov, A.A., Sturm–Liouville operators with singular potentials, Math. Notes, 1999, vol. 66, no. 6, pp. 741–753.
Savchuk, A.M. and Shkalikov, A.A., Sturm–Liouville operator with distribution potentials, Trans. Moscow Math. Soc., 2003, vol. 2003, pp. 143–192.
Eckhardt, J., Gesztesy, F., Nichols, R., and Teschl, G., Weyl–Titchmarsh theory for Sturm–Liouville operators with distributional potentials, Opuscula Math., 2013, vol. 33, no. 3, pp. 467–563.
Kanguzhin, B.E. and Tokmagambetov, N.E., Resolvents of well-posed problems for finite-rank perturbations of the polyharmonic operator in a punctured domain, Sib. Math. J., 2016, vol. 57, no. 2, pp. 265–273.
Sadovnichii, V.A. and Kanguzhin, B.E., A connection between the spectrum of a differential operator with symmetric coefficients and the boundary conditions, Dokl. Math., 1982, vol. 26, pp. 614–618.
Savchuk, A.M. and Shkalikov, A.A., Trace formula for Sturm–Liouville operators with singular potentials, Math. Notes, 2001, vol. 69, no. 3, pp. 387–400.
Kanguzhin, B.E. and Tokmagambetov, N.E., On regularized trace formulas for a well-posed perturbation of the m-Laplace operator, Differ. Equations, 2015, vol. 51, no. 12, pp. 1583–1588.
Kanguzhin, B.E. and Tokmagambetov, N.E., A regularized trace formula for a well-perturbed Laplace operator, Dokl. Math., 2015, vol. 91, no. 1, pp. 1–4.
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Original Russian Text © B.E. Kanguzhin, 2018, published in Differentsial’nye Uravneniya, 2018, Vol. 54, No. 1, pp. 9–14.
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Kanguzhin, B.E. Weinstein Criteria and Regularized Traces in the Case of Transverse Vibrations of an Elastic String with Springs. Diff Equat 54, 7–12 (2018). https://doi.org/10.1134/S0012266118010020
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DOI: https://doi.org/10.1134/S0012266118010020