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On spectral asymptotics of the Sturm-Liouville problem with self-conformal singular weight. (English. Russian original) Zbl 07253595

Sib. Math. J. 61, No. 5, 901-912 (2020); translation from Sib. Mat. Zh. 61, No. 5, 1130-1143 (2020).
Summary: Under study is the spectral asymptotics of the Sturm-Liouville problem with a singular self-conformal weight measure. We assume that the conformal iterated function system generating the weight measure satisfies a stronger version of the bounded distortion property. The power exponent of the main term of the eigenvalue counting function asymptotics is obtained under the assumption. This generalizes the result by Fujita in the case of self-similar (self-affine) measures.

MSC:

47Axx General theory of linear operators
47Bxx Special classes of linear operators
47Gxx Integral, integro-differential, and pseudodifferential operators
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[1] Birman, MS; Solomyak, MZ, Spectral Theory of Self-Adjoint Operators in Hilbert Space (1987), Dordrecht: Reidel Publ., Dordrecht
[2] Krein, MG, Determination of the density of the symmetric inhomogeneous string by spectrum, Dokl. Akad. Nauk SSSR, 76, 3, 345-348 (1951) · Zbl 0042.09502
[3] Birman, MS; Solomyak, MZ, Asymptotic behavior of the spectrum of weakly polar integral operators, Math. USSR-Izv., 4, 5, 1151-1168 (1970) · Zbl 0261.47027 · doi:10.1070/IM1970v004n05ABEH000948
[4] Kac, IS; Krein, MG, A discreteness criterion for the spectrum of a singular string, Izv. Vuzov. Matematika, 2, 136-153 (1958) · Zbl 1469.34111
[5] McKean, HP; Ray, DB, Spectral distribution of a differential operator, Duke Math. J., 29, 2, 281-292 (1962) · Zbl 0114.04902 · doi:10.1215/S0012-7094-62-02928-9
[6] Borzov, VV, The quantitative characteristics of singular measures, Topics in Math. Phys., 4, 37-42 (1971)
[7] Fujita T., “A fractional dimension, self-similarity and a generalized diffusion operator,” Taniguchi Symposium. PMMP. Katata, 83-90 (1985). · Zbl 0652.60084
[8] Freiberg, UR, Refinement of the spectral asymptotics of generalized Krein-Feller operators, Forum Math., 23, 2, 427-445 (2011) · Zbl 1211.35211 · doi:10.1515/form.2011.017
[9] Solomyak, M.; Verbitsky, E., On a spectral problem related to self-similar measures, Bul. London Math. Soc., 27, 3, 242-248 (1995) · Zbl 0823.34071 · doi:10.1112/blms/27.3.242
[10] Kigami, J.; Lapidus, ML, Weyl’s problem for the spectral distributions of Laplacians on p.c.f. self-similar fractals, Commun. Math. Phys., 158, 93-125 (1991) · Zbl 0806.35130 · doi:10.1007/BF02097233
[11] Nazarov, AI, Logarithmic \(L_2 \)-small ball asymptotics with respect to a self-similar measure for some Gaussian processes, J. Math. Sci. (New York), 113, 3, 1314-1327 (2006) · doi:10.1007/s10958-006-0041-x
[12] Freiberg, UR, A survey on measure geometric Laplacians on Cantor like sets, Arab. J. Sci. Eng., 28, 1, 189-198 (2003) · Zbl 1271.28001
[13] Vladimirov, AA; Sheipak, IA, On the Neumann problem for the Sturm-Liouville equation with Cantor-type self-similar weight, Funct. Anal. Appl., 47, 4, 261-270 (2013) · Zbl 1310.34038 · doi:10.1007/s10688-013-0033-9
[14] Rastegaev, NV, On spectral asymptotics of the Neumann problem for the Sturm-Liouville equation with self-similar weight of generalized Cantor type, J. Math. Sci. (New York), 210, 6, 814-821 (2015) · Zbl 1334.34186 · doi:10.1007/s10958-015-2592-1
[15] Rastegaev, NV, On spectral asymptotics of the Neumann problem for the Sturm-Liouville equation with arithmetically self-similar weight of a generalized Cantor type, Funct. Anal. Appl., 52, 1, 70-73 (2018) · Zbl 1394.34183 · doi:10.1007/s10688-018-0211-x
[16] Vladimirov, AA; Sheipak, IA, Indefinite Sturm-Liouville problem for some classes of self-similar singular weights, Proc. Steklov Inst. Math., 255, 82-91 (2006) · Zbl 1351.34023 · doi:10.1134/S0081543806040079
[17] Vladimirov, AA; Sheipak, IA, Self-similar functions in \((L_2[0,1])\) and the Sturm-Liouville problem with singular indefinite weight, Sb. Math., 197, 11, 1569-1586 (2006) · Zbl 1177.34039 · doi:10.1070/SM2006v197n11ABEH003813
[18] Vladimirov, AA, Calculating the eigenvalues of the Sturm-Liouville problem with a fractal indefinite weight, Comp. Math. Math. Phys., 47, 8, 1295-1300 (2007) · Zbl 07811691 · doi:10.1134/S0965542507080076
[19] Freiberg, UR; Rastegaev, NV, On spectral asymptotics of the Sturm-Liouville problem with self-conformal singular weight with strong bounded distortion property, J. Math. Sci. (New York), 244, 6, 1010-1014 (2020) · Zbl 1453.34036 · doi:10.1007/s10958-020-04671-x
[20] Patzschke, N., Self-conformal multifractal measures, Adv. Appl. Math., 19, 486-513 (1997) · Zbl 0912.28007 · doi:10.1006/aama.1997.0557
[21] Hutchinson, JE, Fractals and self similarity, Indiana Univ. Math. J., 30, 5, 713-747 (1981) · Zbl 0598.28011 · doi:10.1512/iumj.1981.30.30055
[22] Sheipak, IA, On the construction and some properties of self-similar functions in the spaces \(L_p[0,1] \), Math. Notes, 81, 6, 827-839 (2007) · Zbl 1132.28006 · doi:10.1134/S0001434607050306
[23] Zolotarev, VM, Asymptotic behavior of Gaussian measure in \(l_2 \), J. Soviet Math., 35, 2, 2330-2334 (1986) · doi:10.1007/BF01105650
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