Geodesic
Self-similar functions in $L_2[0,1]$ and the
Sbornik. Mathematics, Tome 197 (2006) no. 11, pp. 1569-1586 
A. A. Vladimirov; I. A. Sheipak. Self-similar functions in $L_2[0,1]$ and the. Sbornik. Mathematics, Tome 197 (2006) no. 11, pp. 1569-1586. http://geodesic.mathdoc.fr/item/SM_2006_197_11_a1/
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Voir la notice de l'article provenant de la source Math-Net.Ru

The question of the asymptotic behaviour of the spectrum of the boundary value problem \begin{equation*} -y''-\lambda\rho y=0, \qquad y(0)=y(1)=0, \end{equation*} is considered, where $\rho$ is a function in $\mathring W_2^{-1}[0,1]$ with arithmetically self-similar primitive function. It is not assumed here that the weight $\rho$ has a constant sign. The theoretical results obtained are illustrated by the data of numerical calculations. Bibliography: 10 titles.

[1] B. Ćurgus, H. Langer, “A Krein space approach to symmetric ordinary differential operators with an indefinite weight function”, J. Differential Equations, 79:1 (1989), 31–61 | DOI | MR | Zbl

[2] T. Ya. Azizov, I. S. Iokhvidov, Osnovy teorii lineinykh operatorov v prostranstvakh s indefinitnoi metrikoi, Nauka, M., 1986 | MR | Zbl

[3] I. S. Kats, M. G. Krein, “O spektralnykh funktsiyakh struny”, v kn. F. Atkinson, Diskretnye i nepreryvnye granichnye zadachi, Mir, M., 1968, 648–733 | MR | MR | Zbl

[4] M. Solomyak, E. Verbitsky, “On a spectral problem related to self-similar measures”, Bull. London Math. Soc., 27:3 (1995), 242–248 | DOI | MR | Zbl

[5] A. M. Savchuk, A. A. Shkalikov, “Operatory Shturma–Liuvillya s potentsialami-raspredeleniyami”, Tr. MMO, 64, 2003, 159–212 | MR | Zbl

[6] M. G. Krein, “Vvedenie v geometriyu indefinitnykh $J$-prostranstv i teoriyu operatorov v etikh prostranstvakh”, Vtoraya letnyaya matematicheskaya shkola, t. 1, Naukova dumka, Kiev, 1965, 15–92 | MR | Zbl

[7] P. Lancaster, A. Shkalikov, Qiang Ye, “Strongly definitizable linear pencils in Hilbert space”, Integral Equations Operator Theory, 17:3 (1993), 338–360 | DOI | MR | Zbl

[8] M. Levitin, D. Vassiliev, “Spectral asymptotics, renewal theorem, and the Berry conjecture for a class of fractals”, Proc. London Math. Soc. (3), 72 (1996), 188–214 | DOI | MR | Zbl

[9] V. Feller, Vvedenie v teoriyu veroyatnostei i ee prilozheniya, t. 2, Mir, M., 1967 | MR | MR | Zbl

[10] F. Riss, B. Sekefalvi-Nad, Lektsii po funktsionalnomu analizu, Mir, M., 1979 | MR | MR | Zbl