Uniqueness of recovery of the Sturm-Liouville operator with a spectral parameter quadratically entering the boundary condition

L. I. Mammadova   ; I. M. Nabiev

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 79 (2022), pp. 14-24

Résumé

The work is devoted to the study of the inverse problem for the Sturm-Liouville operator with a real square-integrable potential. The boundary conditions are non-separated. One of these boundary conditions includes a quadratic function of the spectral parameter. A uniqueness theorem is proved and an algorithm for solving the inverse problem is constructed. As spectral data, we use the spectrum of the considered boundary value problem, the constant term of the quadratic function of the spectral parameter included in the boundary condition, and some special sequence of signs. From these spectral data, the characteristic function of the boundary value problem is first reconstructed in the form of an infinite product and the parameters of the boundary conditions, and then the problem is reduced to the inverse problem of reconstructing the potential of the Sturm-Liouville operator from the spectra of two boundary value problems with separated boundary conditions. The results of the article can be used for solving various versions of inverse problems of spectral analysis for differential operators, as well as for integrating some nonlinear equations of mathematical physics.

Détail

Keywords: Sturm-Liouville operator, nonseparated boundary conditions, inverse problem, uniqueness theorem
Mots-clés : solution algorithm.

Bibliographie

[1] Kollatts L., Zadachi na sobstvennye znacheniya : (s tekhnicheskimi prilozheniyami), Nauka, M., 1968, 504 pp.

[2] Tikhonov A.N., Samarskii A.A., Uravneniya matematicheskoi fiziki, Izd-vo Mosk. gos. un-ta, M., 1999, 799 pp.

[3] Akhtyamov A.M., Teoriya identifikatsii kraevykh uslovii i ee prilozheniya, Fizmatlit, M., 2009, 272 pp.

[4] Panakhov E.S., Koyunbakan H., Ic U., “Reconstruction formula for the potential function of Sturm-Liouville problem with eigenparameter boundary condition”, Inverse Probl. Sci. and Eng., 18:1 (2010), 173–180 | DOI

[5] Etkin A.E., Etkina G.P., “O edinstvennosti resheniya obratnoi zadachi Shturma-Liuvillya so spektralnym parametrom, ratsionalno vkhodyaschim v granichnoe uslovie”, Izvestiya Irkutskogo gosudarstvennogo universiteta. Ser. Matematika, 4:3 (2011), 158170

[6] Guldu Y., Amirov R.Kh., Topsakal N., “On impulsive Sturm-Liouville operators with singularity and spectral parameter in boundary conditions”, Ukrainskii matematicheskii zhurnal, 64:12 (2012), 1610–1629

[7] Moller M., Pivovarchik V., Spectral Theory of Operator Pencils, Hermite-Biehler Functions, and their Applications, Birkhauser, Cham, 2015, 412 pp. | DOI

[8] Guliyev N.J., “Schrodinger operators with distributional potentials and boundary conditions dependent on the eigenvalue parameter”, j. Math. Phys., 60:6 (2019), 063501, 1–23 | DOI

[9] Guliyev N.J., “On two-spectra inverse problems”, Proc. American Math. Soc., 148:10 (2020), 4491–4502 | DOI

[10] Guliyev N.J., “Essentially isospectral transformations and their applications”, Annali di Matematica Pura ed Applicata, 199:4 (2020), 1621–1648 | DOI

[11] Ala V., Mamedov Kh.R., “On a discontinuous Sturm-Liouville problem with eigenvalue parameter in the boundary conditions”, Dynamic Systems and Applications, 29 (2020), 182–191 http://www.dynamicpublishers.com/DSA/dsa2020pdf/11-DSA-20-A-11.pdf

[12] Yang Ch.-F., Bondarenko N.P., Xu X-Ch., “An inverse problem for the Sturm Liouville pencil with arbitrary entire functions in the boundary condition”, Inverse Problems and Imaging, 14:1 (2020), 153–169 | DOI

[13] Sadovnichii V.A., Sultanaev Ya.T., Akhtyamov A.M., “Obratnaya zadacha dlya puchka operatorov s neraspadayuschimisya kraevymi usloviyami”, Doklady RAN, 425:1 (2009), 31–33

[14] Yurko V.A., “Inverse problems for nonselfadjoint quasi-periodic differential pencils”, Anal. Math. Phys., 2 (2012), 215–230 | DOI

[15] Freiling G., Yurko V., “Recovering nonselfadjoint differential pencils with nonseparated boundary conditions”, Applicable Anal., 94:8 (2015), 1649–1661 | DOI

[16] Ibadzadeh Ch.G., Mammadova L.I., Nabiev I.M., “Inverse problem of spectral analysis for diffusion operator with nonseparated boundary conditions and spectral parameter in boundary condition”, Azerbaijan Journal of Mathematics, 9:1 (2019), 171–189 http://azjm.org/volumes/0901/pdf/1.pdf

[17] Mammadova L.I., Nabiev I.M., Rzayeva Ch.H., “Uniqueness of the solution of the inverse problem for differential operator with semiseparated boundary conditions”, Baku Mathematical Journal, 1:1 (2022), 47–52 | DOI

[18] Nabiev I.M., “Reconstruction of the differential operator with spectral parameter in the boundary condition”, Mediterr. Journal of Mathematics, 19:3 (2022), 124, 1–14 | DOI

[19] Yurko V.A., “Inverse spectral problems for differential operators with non-separated boundary conditions”, Journal of Inverse and Ill-posed Problems, 28:4 (2020), 567–616 | DOI

[20] Marchenko V.A., Operatory Shturma-Liuvillya i ikh prilozheniya, Naukova dumka, Kiev, 1977, 332 pp.

[21] Nabiev I.M., “Determination of the diffusion operator on an interval”, Colloquium Mathematicum, 134:2 (2014), 165–178 | DOI

[22] Yurko V.A., “Ob obratnoi periodicheskoi zadache dlya tsentralno-simmetrichnykh potentsialov”, Izvestiya Saratovskogo universiteta. Novaya seriya. Ser. Matematika. Mekhanika. Informatika, 16:1 (2016), 68–75 | DOI

[23] Guseinov I.M., Nabiev I.M., “Reshenie odnogo klassa obratnykh kraevykh zadach Shturma-Liuvillya”, Matematicheskii sbornik, 186:5 (1995), 35–48 | DOI

[24] Makin A.S., “Obratnaya zadacha dlya operatora Shturma-Liuvillya s regulyarnymi kraevymi usloviyami”, Doklady RAN, 408:3 (2006), 305–308

[25] Levin B.Ya., Tselye funktsii, Izd-vo Mosk. gos. un-ta, M., 1971, 124 pp.

[26] Mammadova L.I., Nabiev I.M., “Spektralnye svoistva operatora Shturma-Liuvillya so spektralnym parametrom, kvadratichno vkhodyaschim v granichnoe uslovie”, Vestnik Udmurtskogo universiteta. Matematika. Mekhanika. Kompyuternye nauki, 30:2 (2020), 237–248 | DOI