Abstract
The Sturm–Liouville problem on a finite closed interval with potential and weight of first order of singularity is studied. Estimates for the s-numbers and eigenvalues of the corresponding integral operator are obtained. The spectral trace of first negative order is evaluated in terms of the integral kernel. The obtained theoretical results are illustrated by examples.
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References
M. G. Krein, “On a generalization of investigations of Stieltjes,” Dokl. Akad. Nauk SSSR 87 (6), 881–884 (1952).
M. G. Krein and I. S. Kats, “On the spectral functions of the string,” Amer. Math. Soc. Transl. 103 (2), 19–102 (1974).
W. Feller, “Generalized second-order differential operators and their lateral conditions,” Illinois J. Math. 1, 459–504 (1957).
Y. Kasahara, “Spectral theory of generalized second-order differential operators and its applications to Markov processes,” Japan J. Math. (N. S.) 1 (1), 67–84 (1975).
H. Dym and H. P. McKean, Gaussian Processes, Function Theory, and the Inverse Spectral Problem, in Probab. and Math. Stat. (Academic Press, New York, 1976), Vol. 31.
I. S. Kats, “The spectral theory of a string,” Ukrainian Math. J. 46 (3), 159–182 (1994).
S. Kotani and S. Watanabe, “Krein’s spectral theory of strings and generalized diffusion processes,” in Functional Analysis in Markov Processes, Lecture Notes in Math. (Springer-Verlag, Berlin, 1982), Vol. 923, pp. 235–259.
A. Fleige, Spectral Theory of Indefinite Krein–Feller Differential Operators, in Math. Res. (Akademie Verlag, Berlin, 1996), Vol. 98.
A. Zettl, Sturm–Liouville Theory, in Math. Surveys Monogr. (Amer. Math. Soc., Providence, RI, 2005), Vol. 121.
J. Eckhardt and A. Kostenko, “The inverse spectral problem for indefinite strings,” Invent. Math. 204 (3), 939–977 (2016).
A. Constantin, V. S. Gerdjikov, and R. I. Ivanov, “Inverse scattering transform for the Camassa–Holm equation,” Inverse Problems 22 (6), 2197–2207 (2006).
C. Bennewitz, B. M. Brown, and R. Weikard, “Scattering and inverse scattering for a left-definite Sturm–Liouville problem,” J. Differential Equations 253 (8), 2380–2419 (2012).
J. Eckhardt and A. Kostenko, “An isospectral problem for global conservative multi-peakon solutions of the Camassa–Holm equation,” Comm. Math. Phys. 329 (3), 893–918 (2014).
A. M. Savchuk and A. A. Shkalikov, “Sturm-Liouville operators with singular potentials,” Mat. Zametki 66 (6), 897–912 (1999) [Math. Notes 66 (5–6), 741–753 (2000)].
A. M. Savchuk and A. A. Shkalikov, “Sturm-Liouville operators with distribution potentials”, with the potentials distributions,” Trudy Moskov. Mat. Obshch. 64, 159–212 (2003) [Trans. Moscow Math. Soc. 64, 143–192 (2003)].
A. A. Vladimirov and I. A. Sheipak, “Self-similar functions in the space L2[0, 1] and the Sturm-Liouville problem with a singular indefinite weight,” Mat. Sb. 197 (11), 13–30 (2006) [Sb. Math. 197 (11–12), 1569–1586 (2006)].
A. A. Vladimirov and I. A. Sheipak, “The indefinite Sturm-Liouville problem for some classes of self-similar singular weights,” in Trudy Mat. Inst. Steklov Vol. 255: Function Spaces, Approximation Theory, Nonlinear Analysis (Nauka, Moscow, 2006), pp. 88–98 [Proc. Steklov Inst. Math. 255 (4), 82–91 (2006)].
I. A. Sheipak, “On the construction and some properties of self-similar functions in the spaces L p[0, 1],” Mat. Zametki 81 (6), 924–938 (2007) [Math. Notes 81 (5–6), 827–839 (2007)].
A. A. Vladimirov and I. A. Sheipak, “Asymptotics of the eigenvalues of the Sturm-Liouville problem with discrete self-similar weight,” Mat. Zametki 88 (5), 662–672 (2010) [Math. Notes 88 (5–6), 637–646 (2010)].
A. A. Vladimirov and I. A. Sheipak, “On the Neumann problem for the Sturm-Liouville equation with Cantor-type self-similar weight,” Funktsional. Anal. Prilozhen. 47 (4), 18–29 (2013) [Functional Anal. Appl. 47 (4), 261–270 (2013)].
M. Solomyak and E. Verbitsky, “On a spectral problem related to self-similarmeasures,” Bull. London Math. Soc. 27 (3), 242–248 (1995).
A. I. Nazarov, “Logarithmic asymptotics of small deviations for some Gaussian processes in the L 2-normwith respect to a self-similarmeasure,” in Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) (POMI, St. Petersburg., 2004), Vol. 311, pp. 190–213 [J. Math. Sc. (New York) 133 (3), 1314–1327 (2006)].
A. A. Vladimirov, “Some remarks on the integral characteristics of the Wiener process,” Dal’nevost. Mat. Zh. 15 (2), 156–165 (2015).
V. A. Sadovnichii and V. E. Podol’skii, “Traces of operators,” Uspekhi Mat. Nauk 61 (5 (371)), 89–156 (2006) [Russian Math. Surveys 61 (5), 885–953 (2006)].
T. Kato, Perturbation Theory for Linear Operators (Springer-Verlag, Heidelberg, 1966; Mir, Moscow, 1972).
I. Ts. Gokhberg and M. G. Krein, Introduction to the Theory of Linear Non–Self-Adjoint Operators in Hilbert Space (Nauka, Moscow, 1965) [in Russian].
S. G. Krein, Yu. I. Petunin, and E. M. Semenov, Interpolation of Linear Operators (Nauka, Moscow, 1978) [in Russian].
Ph. Hartman, Ordinary Differential Equations (John Wiley, New York–London–Sydney, 1964; Mir, Moscow, 1970).
A. A. Vladimirov, “On the oscillation theory of the Sturm-Liouville problem with singular coefficients,” Zh. Vychisl. Mat. Mat. Fiz. 49 (9), 1609–1621 (2009) [Comput. Math. Math. Phys. 49 (9), 1535–1546 (2009)].
H. Triebel, Interpolation Theory, Function Spaces, Differential Operators (Birkhäuser, Berlin, 1977; Mir, Moscow, 1980).
R. Oloff, “Interpolation zwischen den Klassen Sp von Operatoren in Hilberträumen,” Math. Nachr. 46, 209–218 (1970).
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Original Russian Text © A. S. Ivanov, A. M. Savchuk, 2017, published in Matematicheskie Zametki, 2017, Vol. 102, No. 2, pp. 197–215.
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Ivanov, A.S., Savchuk, A.M. Trace of order (−1) for a string with singular weight. Math Notes 102, 164–180 (2017). https://doi.org/10.1134/S0001434617070197
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DOI: https://doi.org/10.1134/S0001434617070197