Weighted string equation where the weight is a noncompact multiplier: continuous spectrum and eigenvalues

Sharov, E. B.; Sheipak, I.

doi:10.1090/spmj/1723

Weighted string equation where the weight is a noncompact multiplier: continuous spectrum and eigenvalues
HTML articles powered by AMS MathViewer

by E. B. Sharov and I. A. Sheipak
Translated by: S. V. Kislyakov

St. Petersburg Math. J. 33 (2022), 697-709

DOI: https://doi.org/10.1090/spmj/1723

Published electronically: June 27, 2022

PDF | Request permission

Abstract:

The oscillation equation for a singular string with discrete weight generated by a self-similar $n$-link multiplier in the Sobolev space with a negative smoothness index is considered. It is shown that in the case of a noncompact multiplier, the string problem is equivalent to the spectral problem for an $(n-1)$-periodic Jacobi matrix. In the case of $n=3$, a complete description of the spectrum of the problem is given, and a criterion for emergence of an eigenvalue in a gap of the continuous spectrum is obtained.

References

M. G. Kreĭn, Determination of the density of a nonhomogeneous symmetric cord by its frequency spectrum, Doklady Akad. Nauk SSSR (N.S.) 76 (1951), 345–348 (Russian). MR 0042045
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. 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
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. A. Vladimirov and I. A. Sheĭpak, The indefinite Sturm-Liouville problem for some classes of self-similar singular weights, Tr. Mat. Inst. Steklova 255 (2006), no. Funkts. Prostran., Teor. Priblizh., Nelineĭn. Anal., 88–98 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 4(255) (2006), 82–91. MR 2301611, DOI 10.1134/s0081543806040079
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
Yu. V. Tikhonov and I. A. Sheĭpak, On the string equation with a singular weight belonging to the space of multipliers in Sobolev spaces with negative index of smoothness, Izv. Ross. Akad. Nauk Ser. Mat. 80 (2016), no. 6, 258–273 (Russian, with Russian summary); English transl., Izv. Math. 80 (2016), no. 6, 1242–1256. MR 3588822, DOI 10.4213/im8388
I. S. Kac and M. G. Kreĭn, Criteria for the discreteness of the spectrum of a singular string, Izv. Vysš. Učebn. Zaved. Matematika 1958 (1958), no. 2 (3), 136–153 (Russian). MR 0139804
Yu. V. Tikhonov, The definition of a self-similar function in quasi-Banach spaces, Mat. Zametki 106 (2019), no. 6, 917–923 (Russian, with Russian summary); English transl., Math. Notes 106 (2019), no. 5-6, 980–985. MR 4045677, DOI 10.4213/mzm12509
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
I. A. Sheĭpak, On the spectrum of the Jacobi operator with exponentially growing matrix elements, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 6 (2011), 15–21 (Russian, with English and Russian summaries); English transl., Moscow Univ. Math. Bull. 66 (2011), no. 6, 244–249. MR 2964319, DOI 10.3103/S0027132211060040
I. V. Stankevič, A certain inverse problem for periodic Jacobi matrices, Mat. Zametki 8 (1970), 297–307 (Russian). MR 276812
N. I. Akhiezer and I. M. Glazman, Theory of linear operators in Hilbert space. Vol. I, Monographs and Studies in Mathematics, vol. 9, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1981. Translated from the third Russian edition by E. R. Dawson; Translation edited by W. N. Everitt. MR 615736
N. I. Akhiezer, The classical moment problem and some related questions in analysis, Hafner Publishing Co., New York, 1965. Translated by N. Kemmer. MR 0184042
E. D. Hellinger and H. S. Wall, Contributions to the analytic theory of continued fractions and infinite matrices, Ann. of Math. (2) 44 (1943), 103–127. MR 8102, DOI 10.2307/1969069