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 Mat. Sb., 1998, Volume 189, Number 1, Pages 133–148 (Mi msb298)

A class of Sturm–Liouville operators and approximate calculation of the first eigenvalues

V. A. Sadovnichii, V. E. Podolskii

M. V. Lomonosov Moscow State University

Abstract: A special class $S$ of Sturm–Liouville operators with simple asymptotic properties of eigenfunctions is investigated. The analytic properties of the potentials are analyzed and the operators in this class are described in terms of the transition function of the inverse problem. The following result is established: the class $S$ is dense in the set of Sturm–Liouville operators with potentials in $L_2$. A subset of $S$ that also has the density property is effectively distinguished. Based on the properties of the operators in this subset, a method of the approximate evaluation of the first eigenvalues of a Sturm–Liouville operator through its regularized traces is proposed and substantiated.

DOI: https://doi.org/10.4213/sm298

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English version:
Sbornik: Mathematics, 1998, 189:1, 129–145

Bibliographic databases:

UDC: 517.94
MSC: 34B24, 34L15

Citation: V. A. Sadovnichii, V. E. Podolskii, “A class of Sturm–Liouville operators and approximate calculation of the first eigenvalues”, Mat. Sb., 189:1 (1998), 133–148; Sb. Math., 189:1 (1998), 129–145

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb298
• https://doi.org/10.4213/sm298
• http://mi.mathnet.ru/eng/msb/v189/i1/p133

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Sadovnichii V.A., Dubrovskii V.V., Maleko E.M., “Ob odnom sposobe priblizhennogo nakhozhdeniya sobstvennykh chisel operatora Shturma-Liuvillya”, Dokl. RAN, 369:1 (1999), 16–18
2. Sadovnichii, VA, “An estimate for the best approximation of solutions of the Sturm-Liouville problem with an analytic potential by partial sums of asymptotic series”, Differential Equations, 35:4 (1999), 498
3. Sadovnichii, VA, “Estimates for the coefficients of asymptotic series for the solutions of the Sturm-Liouville equation with an analytic potential. I”, Differential Equations, 35:2 (1999), 284
4. Sadovnichii, AVA, “Nonuniqueness of solutions to the regularized trace system”, Doklady Mathematics, 71:3 (2005), 411
5. V. A. Sadovnichii, V. E. Podolskii, “Traces of operators”, Russian Math. Surveys, 61:5 (2006), 885–953
6. Sadovnichii V.A.. Podol'skii V.E., “Traces of differential operators”, Differ. Equ., 45:4 (2009), 477–493
7. Aslanova N.M., “Study of the asymptotic eigenvalue distribution and trace formula of a second order operator-differential equation”, Bound Value Probl, 2011, 7
8. Maleko E.M., “O metode sledov rezolvent, vychislennykh tochno”, Vestnik Samarskogo gosudarstvennogo universiteta, 2011, no. 86, 37–52
9. M. K. Kerimov, “Approximate computation of eigenvalues and eigenfunctions of Sturm–Liouville differential operators by applying the theory of regularized traces”, Comput. Math. Math. Phys., 52:3 (2012), 351–386
10. Intissar A., “Regularized Trace Formula of Magic Gribov Operator on Bargmann Space”, J. Math. Anal. Appl., 437:1 (2016), 59–70
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