On spectral asymptotics of the Neumann problem for the Sturm-Liouville equation with self-similar weight of generalized Cantor type. (English. Russian original) Zbl 1334.34186
J. Math. Sci., New York 210, No. 6, 814-821 (2015); translation from Zap. Nauchn. Semin. POMI 425, 86-98 (2014).
Summary: Spectral asymptotics of the weighted Neumann problem for the Sturm-Liouville equation is considered. The weight is assumed to be the distributional derivative of a self-similar generalized Cantor type function. The spectrum is shown to have a periodicity property for a wide class of Cantor type self-similar functions. A weaker “quasiperiodicity” property is established under certain mixed boundary-value conditions. This allows for a more precise description of the main term of the eigenvalue counting function asymptotics.
MSC:
| 34L20 | Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators |
| 34B09 | Boundary eigenvalue problems for ordinary differential equations |
| 34L05 | General spectral theory of ordinary differential operators |
References:
| [1] | M. Solomyak and E. Verbitsky, “On a spectral problem related to self-similar measures,” Bull. London Math. Soc., 27, No. 3, 242-248 (1995). · Zbl 0823.34071 · doi:10.1112/blms/27.3.242 |
| [2] | A. A. Vladimirov and I. A. Sheipak, “On the Neumann problem for the Sturm-Liouville equation with Cantor-type self-similar weight,” Funct. Anal. Its Appl., 47, No. 4, 261-270 (2013). · Zbl 1310.34038 · doi:10.1007/s10688-013-0033-9 |
| [3] | M. Sh. Birman and M. Z. Solomyak, Spectral Theory of Self-Adjoint Operators in Hilbert Space, [in Russian], 2th ed., Lan’ publishers (2010). · Zbl 0098.06002 |
| [4] | M. G. Krein, “Determination of the density of the symmetric inhomogeneous string by spectrum,” Dokl. Akad. Nauk SSSR, 76, No. 3, 345-348 (1951). · Zbl 0042.09502 |
| [5] | M. Sh. Birman and M. Z. Solomyak, “Asymptotic behavior of the spectrum of weakly polar integral operators,” Izv. Math. Nauk, 4, No. 5, 1151-1168 (1970). · Zbl 0261.47027 · doi:10.1070/IM1970v004n05ABEH000948 |
| [6] | V. V. Borzov, “On the quantitative characteristics of singular measures,” Problems Math. Phys., 4, 42-47 (1970). · Zbl 0226.28003 |
| [7] | T. Fujita, “A fractional dimention, self-similarity and a generalized diffusion operator,” in: Taniguchi Symp. PMMP, Katata (1985), pp. 83-90. · Zbl 0261.47027 |
| [8] | I. Hong and T. Uno, “Some consideration of asymptotic distribution of eigenvalues for the equation <Emphasis Type=”Italic“>d2 <Emphasis Type=”Italic“>u/dx2 + <Emphasis Type=”Italic“>λρ(<Emphasis Type=”Italic“>x)<Emphasis Type=”Italic“>u = 0,” Japan. J. Math., 29, 152-164 (1959). · Zbl 0098.06002 |
| [9] | H. P. McKean and D. B. Ray, “Spectral distribution of a differential operator,” Duke Math. J., 29, 281-292 (1962). · Zbl 0114.04902 · doi:10.1215/S0012-7094-62-02928-9 |
| [10] | J. Kigami amd M. L. Lapidus, “Weyls problem for the spectral distributions of Laplacians on p.c.f. self-similar fractals,” Comm. Math. Phys., 158, 93-125 (1991). · Zbl 0806.35130 · doi:10.1007/BF02097233 |
| [11] | A. I. Nazarov, “Logarithmic <Emphasis Type=”Italic“>L2-small ball asymptotics with respect to self-similar measure for some Gaussian processes,” J. Math. Sci., 133, No. 3, 1314-1327 (2006). · doi:10.1007/s10958-006-0041-x |
| [12] | A. A. Vladimirov and I. A. Sheipak, “Self-similar functions in <Emphasis Type=”Italic“>L2[0<Emphasis Type=”Italic“>, 1] and the Sturm- Liouville problem with singular indefinite weight,” Mat. Sb., 197, No. 11, 1569-1586, (2006). · Zbl 1177.34039 · doi:10.1070/SM2006v197n11ABEH003813 |
| [13] | A. A. Vladimirov, “Method of oscillation and spectral problem for four-order differential operator with self-similar weight,” arXiv:1107.4791. · Zbl 1343.34193 |
| [14] | I. A. Sheipak, “On the construction and some properties of self-similar functions in the spaces <Emphasis Type=”Italic“>L <Emphasis Type=”Italic“>p[0<Emphasis Type=”Italic“>, 1],” Math. Notes, 81, No. 6, 827-839 (2007). · Zbl 1132.28006 · doi:10.1134/S0001434607050306 |
| [15] | J. E. Hutchinson, “Fractals and self similarity,” Indiana Univ. Math. J., 30, No. 5, 713-747 (1981). · Zbl 0598.28011 · doi:10.1512/iumj.1981.30.30055 |
| [16] | A. A. Vladimirov, “On the oscillation theory of the Sturm-Liouville problem with singular coefficients,” Comp. Math. Math. Physics, 49, No. 9, 1535-1546 (2009). · Zbl 1199.34136 · doi:10.1134/S0965542509090085 |
| [17] | A. I. Nazarov, “On a set of transformations of Gaussian random functions,” Theory Probab. Appl., 54, No. 2, 203-216 (2010). · Zbl 1214.60011 · doi:10.1137/S0040585X97984103 |
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