Asymptotics of eigenvalues in a problem of high even order with discrete self-similar weight
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A. A. Vladimirov and I. A. Sheipak
Translated by: S. V. Kislyakov - St. Petersburg Math. J. 24 (2013), 263-273
- DOI: https://doi.org/10.1090/S1061-0022-2013-01237-4
- Published electronically: January 22, 2013
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Abstract:
Spectral asymptotics for the boundary problem $(-1)^n y^{(2n)}-\lambda \rho y=0$, $y^{(k)}(0)=y^{(k)}(1)=0$, $0\leq k<n$, is studied in the case where the order $2n$ of the equation satisfies the inequality $n>1$, and the weight $\rho \in W_2^{-1}[0,1]$ is the generalized derivative of a self-similar function $P\in L_2[0,1]$ of zero spectral order.References
- A. A. Vladimirov and I. A. Sheĭpak, Asymptotics of the eigenvalues of the Sturm-Liouville problem with discrete self-similar weight, Mat. Zametki 88 (2010), no. 5, 662–672 (Russian, with Russian summary); English transl., Math. Notes 88 (2010), no. 5-6, 637–646. MR 2868390, DOI 10.1134/S0001434610110039
- M. A. Naĭmark, Lineĭ nye differentsial′nye operatory, Izdat. “Nauka”, Moscow, 1969 (Russian). Second edition, revised and augmented; With an appendix by V. È. Ljance. MR 0353061
- A. I. Nazarov, Logarithmic asymptotics of small deviations for some Gaussian processes in the $L_2$-norm with respect to a self-similar measure, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 311 (2004), no. Veroyatn. i Stat. 7, 190–213, 301 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 133 (2006), no. 3, 1314–1327. MR 2092208, DOI 10.1007/s10958-006-0041-x
- M. Solomyak and E. Verbitsky, On a spectral problem related to self-similar measures, Bull. London Math. Soc. 27 (1995), no. 3, 242–248. MR 1328700, DOI 10.1112/blms/27.3.242
- A. A. Vladimirov and I. A. Sheĭpak, Self-similar functions in the space $L_2[0,1]$ and the Sturm-Liouville problem with a singular indefinite weight, Mat. Sb. 197 (2006), no. 11, 13–30 (Russian, with Russian summary); English transl., Sb. Math. 197 (2006), no. 11-12, 1569–1586. MR 2437086, DOI 10.1070/SM2006v197n11ABEH003813
- A. I. Nazarov and I. A. Sheipak, Degenerate self-similar measures, spectral asymptotics and small derivations of Gaussian processes, arXiv:1009.1252.
- I. A. Sheĭpak, On the construction and some properties of self-similar functions in the spaces $L_p[0,1]$, Mat. Zametki 81 (2007), no. 6, 924–938 (Russian, with Russian summary); English transl., Math. Notes 81 (2007), no. 5-6, 827–839. MR 2349108, DOI 10.1134/S0001434607050306
- I. A. Sheĭpak, Singular points of a self-similar function of spectral zero order: a self-similar Stieltjes string, Mat. Zametki 88 (2010), no. 2, 303–316 (Russian, with Russian summary); English transl., Math. Notes 88 (2010), no. 1-2, 275–286. MR 2867056, DOI 10.1134/S0001434610070254
- P. Lancaster, A. Shkalikov, and Qiang Ye, Strongly definitizable linear pencils in Hilbert space, Integral Equations Operator Theory 17 (1993), no. 3, 338–360. MR 1237958, DOI 10.1007/BF01200290
- A. A. Vladimirov, Estimates for the number of eigenvalues of selfadjoint operator functions, Mat. Zametki 74 (2003), no. 6, 838–847 (Russian, with Russian summary); English transl., Math. Notes 74 (2003), no. 5-6, 794–802. MR 2054002, DOI 10.1023/B:MATN.0000009015.40046.63
- A. A. Vladimirov, On the computation of the eigenvalues of the Sturm-Liouville problem with a fractal indefinite weight, Zh. Vychisl. Mat. Mat. Fiz. 47 (2007), no. 8, 1350–1355 (Russian, with Russian summary); English transl., Comput. Math. Math. Phys. 47 (2007), no. 8, 1295–1300. MR 2378180, DOI 10.1134/S0965542507080076
- N. A. Šanin, Constructive real numbers and constructive functional spaces, Trudy Mat. Inst. Steklov. 67 (1962), 15–294 (Russian). MR 0156786
- B. A. Kushner, Lektsii po konstruktivnomu matematicheskomu analizu, Monographs in Mathematical Logic and Foundations of Mathematics, Izdat. “Nauka”, Moscow, 1973 (Russian). MR 0379147
Bibliographic Information
- A. A. Vladimirov
- Affiliation: A. A. Dorodnitsyn Computing Center, ul. Vavilova, 40, Moscow 119333, Russia
- Email: vladimi@mech.math.msu.su
- I. A. Sheipak
- Affiliation: Department of Mechanics and Mathematics, M. V. Lomonosov Moscow State University, Leninskie gory, Moscow 119992, Russia
- Email: iasheip@mech.math.msu.su
- Received by editor(s): November 6, 2010
- Published electronically: January 22, 2013
- Additional Notes: The authors were supported by RFBR (grants nos. 10-01-00423, 11-01-12115-ofi-m-2011, and 09-06-00125)
- © Copyright 2013 American Mathematical Society
- Journal: St. Petersburg Math. J. 24 (2013), 263-273
- MSC (2010): Primary 34L20
- DOI: https://doi.org/10.1090/S1061-0022-2013-01237-4
- MathSciNet review: 3013324