Geodesic
Restoration of the polynomial potential in the Sturm-Liouville problem
Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 20 (2018) no. 2, pp. 148-158.

Voir la notice de l'article provenant de la source Math-Net.Ru

The problem of identification of medium elasticity by eigenfrequencies of the string oscillating in this medium is considered. The elasticity is supposed to be some polynomial. A solution method based on the representation of linearly independent solutions of the differential equation in the form of Taylor series by variables $x$ and $\lambda$ is presented. A method is also developed that allows to prove uniqueness or non-uniqueness of reconstructed polynomial elasticity coefficient by a finite number of natural frequencies of string vibrations. The latter method is based on the method of arbitrary constant variation. The examples of the problem solution and of the error estimation for the result are given. It is shown that for the unambiguous identification of $n+1$ coefficients of the $n$th power polynomial, which is a potential in the Sturm-Liouville problem, it is sufficient to use $n+1$ eigenvalue. These eigenvalues are found from two different boundary value problems, that differ in one of the boundary conditions. Only a half of eigenvalues’ number in each problem must be taken into account. If this number is odd, the number of eigenvalues that should be taken from one of the problems’ spectra must be increased by one. A counterexample is given showing that eigenfrequencies taken only from one spectrum don’t allow to find the unique solution. In fact, these results improve the well-known Borg theorem in the case when the potential is a polynomial. Also the method is proposed that helps to find the isospectral class of problems for which the range of frequencies is the same.
Keywords: eigenvalue problem, potential identification, string, inverse problem, eigenvalues. 
@article{SVMO_2018_20_2_a1,
     author = {A. M. Akhtyamov and I. {\CYRM}. Utyashev},
     title = {Restoration of the polynomial potential in the {Sturm-Liouville} problem},
     journal = {\v{Z}urnal Srednevol\v{z}skogo matemati\v{c}eskogo ob\^{s}estva},
     pages = {148--158},
     publisher = {mathdoc},
     volume = {20},
     number = {2},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SVMO_2018_20_2_a1/}
}
TY  - JOUR
AU  - A. M. Akhtyamov
AU  - I. М. Utyashev
TI  - Restoration of the polynomial potential in the Sturm-Liouville problem
JO  - Žurnal Srednevolžskogo matematičeskogo obŝestva
PY  - 2018
SP  - 148
EP  - 158
VL  - 20
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SVMO_2018_20_2_a1/
LA  - ru
ID  - SVMO_2018_20_2_a1
ER  - 
%0 Journal Article
%A A. M. Akhtyamov
%A I. М. Utyashev
%T Restoration of the polynomial potential in the Sturm-Liouville problem
%J Žurnal Srednevolžskogo matematičeskogo obŝestva
%D 2018
%P 148-158
%V 20
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SVMO_2018_20_2_a1/
%G ru
%F SVMO_2018_20_2_a1
A. M. Akhtyamov; I. М. Utyashev. Restoration of the polynomial potential in the Sturm-Liouville problem. Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 20 (2018) no. 2, pp. 148-158. http://geodesic.mathdoc.fr/item/SVMO_2018_20_2_a1/

[1] W. A. Ambarzumijan, “Uber eine Frage der Eigenwerttheorie”, Zeitshrift fur Physik, 1929, no. 53, 690–695 | DOI

[2] G. Borg, “Eine Umkehrung der Sturm– Liouvilleschen Eigenwert Aufgabe. Bestimmung der Differentialgleichung durch die Eigenwarte”, Acta Math, 78:1 (1946), 1–96 | DOI | MR | Zbl

[3] N. Levinson, “The inverse Sturm-Liouville problem”, Math. Tidsskr., 3 (1949), 25–30 | MR

[4] V. A. Marchenko, Operators of Sturm-Liouville and their applications, Naukova dumka, Kiev, 1977, 329 pp. (In Russ.) | MR

[5] B. M. Levitan, Inverse Sturm-Liouville problems and their applications, Nauka Publ., Moscow, 1984, 240 pp. (In Russ.)

[6] A. M. Savchuk, A. A. Shkalikov, “Inverse problems for Sturm-Liouville operators with potentials in Sobolev spaces: Uniform stability”, Functional Analysis and Its Applications, 44:4 (2010), 34–53 (In Russ.) | DOI | MR | Zbl

[7] V. A. Yurko, Introduction to the theory of inverse spectral problems, Fizmatlit Publ., Moscow, 2007, 384 pp. (In Russ.)

[8] I. M. Guseinov, I. M. Nabiev, “The inverse spectral problem for pencils of differential operators”, Mathematics Sbornik, 198:11 (2007), 1579–1598 (In Russ.) | DOI | MR | Zbl

[9] I. M. Nabiev, “The inverse quasiperiodic problem for a diffusion operator”, Doklady Mathematics, 76:1 (2007), 527–529 | MR | Zbl

[10] V. A. Sadovnichii, “The uniqueness of the solution of the inverse problem in the case of a second-order equation with non-decomposable conditions, the regularized sums of a part of the eigenvalues. Factorization of the characteristic determinant”, Soviet Mathematics, 206:2 (1972), 293–296 (In Russ.) | Zbl

[11] V. A. Sadovnichii, Ya. T. Sultanaev, A. M. Akhtyamov, “General inverse Sturm-Liouville problem with symmetric potential”, Azerbaijan Journal of Mathematics, 5:2 (2015), 96-108 | MR | Zbl

[12] M. G. Gasymov, I. M. Guseynov, I. M. Nabiev, “The inverse problem for the Sturm-Liouville operator with non-separated self-adjoint boundary conditions”, Sibirskiy Matematicheskiy zhurnal, 31:6 (1991), 46–54 (In Russ.)

[13] E. L. Korotyaev, D. S. Chelkak, “The inverse Sturm-Liouville problem with mixed boundary conditions”, St. Peterburg Math., 21:5 (2010), 761–778

[14] Kh. R. Mamedov, F. Cetinkaya, “Inverse problem for a class of Sturm-Liouville operator with spectral parameter in boundary condition”, Bound. Value Probl., 2013, 16 pp. http://link.springer.com/journal/volumesAndIssues/13661 | MR | Zbl

[15] E. S. Panakhov, H. Koyunbakan, Ic. Unal, “Reconstruction formula for the potential function of Sturm–Liouville problem with eigenparameter boundary condition”, Inverse Problems in Science and Engineering, 18:1 (2010), 173–180 | DOI | MR | Zbl

[16] A. M. Akhtyamov, A. V. Mouftakhov, “Identification of boundary conditions using natural frequencies”, Inverse Problems in Science and Engineering, 12:4 (2004), 393-408 | DOI | MR

[17] A. M. Akhtyamov, I. M. Utyashev, “Identification of the boundary conditions at both ends of the string according to the natural frequencies of the oscillations”, Akusticheskiy zhurnal, 61:6 (2015), 647-655 (In Russ.) | DOI

[18] I. M. Utyashev, A. M. Akhtyamov, “Identification of the string boundary conditions using natural frequencies”, Trudy Instituta mekhaniki im. R. R. Mavlyutova, 11:1 (2016), 38–52 (In Russ.)

[19] A. Kollatts, Tasks for eigenvalues (with technical applications), Nauka Publ., Moscow, 1968, 504 pp. (In Russ.)

[20] M. A. Naymark, Linear differential operators, Nauka Publ., Moscow, 1969, 526 pp. (In Russ.) | MR