On two-spectra inverse problems
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- by Namig J. Guliyev
- Proc. Amer. Math. Soc. 148 (2020), 4491-4502
- DOI: https://doi.org/10.1090/proc/15155
- Published electronically: July 20, 2020
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Abstract:
We consider a two-spectra inverse problem for the one-dimensional Schrödinger equation with boundary conditions containing rational Herglotz–Nevanlinna functions of the eigenvalue parameter and provide a complete solution of this problem.References
- R. Kh. Amirov, A. S. Ozkan, and B. Keskin, Inverse problems for impulsive Sturm-Liouville operator with spectral parameter linearly contained in boundary conditions, Integral Transforms Spec. Funct. 20 (2009), no. 7-8, 607–618. MR 2543768, DOI 10.1080/10652460902726443
- Agnes Ilona Benedek and Rafael Panzone, On inverse eigenvalue problems for a second-order differential equation with parameter contained in the boundary conditions, Notas de Álgebra y Análisis [Notes on Algebra and Analysis], vol. 9, Universidad Nacional del Sur, Instituto de Matemática, Bahía Blanca, 1980. MR 608412
- Paul A. Binding, Patrick J. Browne, and Bruce A. Watson, Equivalence of inverse Sturm-Liouville problems with boundary conditions rationally dependent on the eigenparameter, J. Math. Anal. Appl. 291 (2004), no. 1, 246–261. MR 2034071, DOI 10.1016/j.jmaa.2003.11.025
- Göran Borg, Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe. Bestimmung der Differentialgleichung durch die Eigenwerte, Acta Math. 78 (1946), 1–96 (German). MR 15185, DOI 10.1007/BF02421600
- M. V. Chugunova, Inverse spectral problem for the Sturm-Liouville operator with eigenvalue parameter dependent boundary conditions, Operator theory, system theory and related topics (Beer-Sheva/Rehovot, 1997) Oper. Theory Adv. Appl., vol. 123, Birkhäuser, Basel, 2001, pp. 187–194. MR 1821912
- Jonathan Eckhardt, Fritz Gesztesy, Roger Nichols, and Gerald Teschl, Inverse spectral theory for Sturm-Liouville operators with distributional potentials, J. Lond. Math. Soc. (2) 88 (2013), no. 3, 801–828. MR 3145132, DOI 10.1112/jlms/jdt041
- Namig J. Guliyev, Inverse eigenvalue problems for Sturm-Liouville equations with spectral parameter linearly contained in one of the boundary conditions, Inverse Problems 21 (2005), no. 4, 1315–1330. MR 2158111, DOI 10.1088/0266-5611/21/4/008
- Namig J. Guliyev, A uniqueness theorem for Sturm-Liouville equations with a spectral parameter linearly contained in the boundary conditions, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb. 25 (2006), 35–40. MR 2332927
- Namig J. Guliyev, Essentially isospectral transformations and their applications, Ann. Mat. Pura Appl. (4) 199 (2020), no. 4, 1621–1648. MR 4117511, DOI 10.1007/s10231-019-00934-w
- Namig J. Guliyev, Schrödinger operators with distributional potentials and boundary conditions dependent on the eigenvalue parameter, J. Math. Phys. 60 (2019), no. 6, 063501, 23. MR 3959138, DOI 10.1063/1.5048692
- Namig J. Guliyev, On extensions of symmetric operators, Oper. Matrices 14 (2020), no. 1, 71–75. MR 4080923, DOI 10.7153/oam-2020-14-05
- Namig J. Guliyev, Inverse square singularities and eigenparameter dependent boundary conditions are two sides of the same coin, submitted. arXiv:2001.00061, 2020.
- I. M. Guseĭnov and I. M. Nabiev, Solution of a class of inverse Sturm-Liouville boundary value problems, Mat. Sb. 186 (1995), no. 5, 35–48 (Russian, with Russian summary); English transl., Sb. Math. 186 (1995), no. 5, 661–674. MR 1341083, DOI 10.1070/SM1995v186n05ABEH000035
- Rostyslav O. Hryniv and Yaroslav V. Mykytyuk, Inverse spectral problems for Sturm-Liouville operators with singular potentials. II. Reconstruction by two spectra, Functional analysis and its applications, North-Holland Math. Stud., vol. 197, Elsevier Sci. B. V., Amsterdam, 2004, pp. 97–114. MR 2098874, DOI 10.1016/S0304-0208(04)80159-2
- Ch. G. Ibadzade and I. M. Nabiev, Reconstruction of the Sturm-Liouville operator with nonseparated boundary conditions and the with a spectral parameter in the boundary condition, Ukraïn. Mat. Zh. 69 (2017), no. 9, 1217–1223 (Russian, with English and Ukrainian summaries); English transl., Ukrainian Math. J. 69 (2018), no. 9, 1416–1423. MR 3722475, DOI 10.1007/s11253-018-1440-0
- M. G. Kreĭn, Solution of the inverse Sturm-Liouville problem, Doklady Akad. Nauk SSSR (N.S.) 76 (1951), 21–24 (Russian). MR 0039895
- N. Dzh. Kuliev, Inverse problems for the Sturm-Liouville equation with a spectral parameter in the boundary condition, Dokl. Nats. Akad. Nauk Azerb. 60 (2004), no. 3-4, 10–16 (Russian, with English and Azerbaijani summaries). MR 2187447
- B. M. Levitan, Determination of a Sturm-Liouville differential equation in terms of two spectra, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 63–78 (Russian). MR 0159980
- B. M. Levitan and M. G. Gasymov, Determination of a differential equation by two spectra, Uspehi Mat. Nauk 19 (1964), no. 2 (116), 3–63 (Russian). MR 0162996
- S. G. Mamedov, Determination of a second-order differential equation from two spectra with a spectral parameter entering into the boundary conditions, Izv. Akad. Nauk Azerbaĭdzhan. SSR Ser. Fiz.-Tekhn. Mat. Nauk 3 (1982), no. 2, 15–22 (Russian, with English and Azerbaijani summaries). MR 687581
- V. A. Marčenko, Some questions of the theory of one-dimensional linear differential operators of the second order. I, Trudy Moskov. Mat. Obšč. 1 (1952), 327–420 (Russian). MR 0058064
- V. A. Marchenko, Operatory Shturma-Liuvillya i ikh prilozheniya, Izdat. “Naukova Dumka”, Kiev, 1977 (Russian). MR 0481179
- V. A. Marčenko and I. V. Ostrovs′kiĭ, A characterization of the spectrum of the Hill operator, Mat. Sb. (N.S.) 97(139) (1975), no. 4(8), 540–606, 633–634 (Russian). MR 0409965
- A. M. Savchuk and A. A. Shkalikov, Inverse problem for Sturm-Liouville operators with distribution potentials: reconstruction from two spectra, Russ. J. Math. Phys. 12 (2005), no. 4, 507–514. MR 2201315
Bibliographic Information
- Namig J. Guliyev
- Affiliation: Institute of Mathematics and Mechanics, Azerbaijan National Academy of Sciences, 9 B. Vahabzadeh str., AZ1141, Baku, Azerbaijan
- MR Author ID: 286202
- ORCID: 0000-0002-5633-6748
- Email: njguliyev@gmail.com
- Received by editor(s): December 16, 2019
- Received by editor(s) in revised form: April 12, 2020
- Published electronically: July 20, 2020
- Communicated by: Tanya Christiansen
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 148 (2020), 4491-4502
- MSC (2010): Primary 34A55, 34B07, 34L40; Secondary 34B24, 47A75, 47E05
- DOI: https://doi.org/10.1090/proc/15155
- MathSciNet review: 4135313