Determination of a differential pencil from interior spectral data on a union of two closed intervals

In this paper, we consider a quadratic pencil of Sturm–Liouville operator on closed sets. We study an interior-inverse problem for this kind operator and give a uniqueness theorem with an appropriate example.

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[1] Adalar I. Hochstadt-Lieberman type theorems for a diffusion operator on a time scale. Quaestiones Mathematicae 2021; doi: 10.2989/16073606.2021.1895351
[2] Adalar I, Ozkan AS. An interior inverse Sturm–Liouville problem on a time scale. Analysis and Mathematical Physics 2020: 10 (4): 1-10.
[3] Bartosiewicz Z, Piotrowska E. Lyapunov functions in stability of nonlinear systems on time scales. Journal of Difference Equations and Applications. 2020; 17 (3): 309-325.
[4] Bohner M, Peterson A. Dynamic Equations on Time Scales. Birkha ̈user, Boston MA, 2001.
[5] Bohner M, Peterson A. Advances in Dynamic Equations on Time Scales. Birkha ̈user, Boston MA, 2003.
[6] Bohner M, Georgiev, SG. Multivariable dynamic calculus on time scales. Cham, Switzerland: Springer, 2016.
[7] Bohner M, Koyunbakan H. Inverse problems for Sturm–Liouville difference equations. Filomat 2016; 30 (5): 1297- 1304.
[8] Bondarenko NP. Inverse problem for the differential pencil on an arbitrary graph with partial information given on the coefficients. Analysis and Mathematical Physics 2019 9 1393–1409.
[9] Buterin SA, Yurko VA. Inverse spectral problem for pencils of differential operators on a finite interval, Vestnik Bashkir. Univ. 2006; 4 8–12.
[10] Buterin SA, Shieh CT. Inverse nodal problem for differential pencils. Applied Mathematics Letters 2009; 22 (8): 1240-1247.
[11] Buterin SA. On half inverse problem for differential pencils with the spectral parameter in boundary conditions. Tamkang journal of mathematics 2011; 42 (3): 355-364.
[12] Buterin SA, Shieh CT. Incomplete inverse spectral and nodal problems for differential pencil, Results Math. 2012; 62 167–179.
[13] Cakmak Y, Isık S. Half Inverse problem for the impulsive diffusion operator with discontinuous coeffcient, Filomat 2016; 30 (1): 157-168.
[14] Freiling G, Yurko VA. Inverse Sturm–Liouville Problems and their Applications. New York, Nova Science,2001.
[15] Gasymov MG, Guseinov GS, Determination of a diffusion operator from the spectral data. Dokl. Akad. Nauk Azerbaijan SSSR 1981; 37 (2): 19-23.
[16] Hilger S. Analysis on measure chains – a unified approach to continuous and discrete calculus. Results in Math. 1990; 18 18–56.
[17] Jaulent M, Jean C. The inverse s-wave scattering problem for a class of potentials depending on energy. Comm. Math. Phys. 1972; 28 177–220.
[18] Koyunbakan H. Inverse problem for a quadratic pencil of Sturm-Liouville operator. J. Math. Anal. Appl. 2011; 378 (2): 549-554.
[19] Kuznetsova MA. A Uniqueness Theorem on Inverse Spectral Problems for the Sturm–Liouville Differential Operators on Time Scales. Results Math 2020; 75 (2): 1-23.
[20] Kuznetsova MA. On Reconstruction of Sturm-Liouville Differential Operators on Time Scales. Mat. notes. 2021; 109 (1): 82-100.
[21] Mochizuki K, Trooshin I. Inverse problem for interior spectral data of Sturm-Liouville operator. J. Inverse Ill-posed Probl. 2001; 9 425-433.
[22] Mosazadeh S. A new approach to uniqueness for inverse Sturm-Liouville problems on finite intervals. Turk. J. Math. 2017; 41 1224-1234.
[23] Neamaty A, Khalili Y. Determination of a differential operator with discontinuity from interior spectral data. Inverse Problems in Science and Engineering 2014; 22 (6): 1002-1008.
[24] Ozkan AS. Sturm-Liouville operator with parameter-dependent boundary conditions on time scales. Electron. J. Differential Equations 2017; 212 1-10.
[25] Ozkan AS. Ambarzumyan-type theorems on a time scale. Journal of Inverse and Ill-posed Problems. 2018 26 (5): 633–637.
[26] Ozkan, AS. Boundary-value problem for a class of second-order parameter-dependent dynamic equations on a time scale. Mathematical Methods in the Applied Sciences 2020; 43 (7): 4353-4359.
[27] Ozkan AS, Adalar I. Half-inverse Sturm-Liouville problem on a time scale. Inverse Problems 2020; 36 025015.
[28] Wang YP, Inverse problems for Sturm–Liouville operators with interior discontinuities and boundary conditions dependent on the spectral parameter. Mathematical Methods in the Applied Sciences 2013; 36 (7): 857-868.
[29] Wang YP. An interior inverse problem for Sturm-Liouville operators with eigenparameter dependent boundary conditions. Tamkang Journal of Mathematics 2011; 42 (3): 395-403.
[30] Wang YP. The inverse problem for differential pencils with eigenparameter dependent boundary conditions from interior spectral data. Applied Mathematics Letters 2012; 25 (7): 1061-1067.
[31] Yang CF, Guo YX. Determination of a differential pencil from interior spectral data. Journal of Mathematical Analysis and Applications 2011; 375 (1): 284-293.
[32] Yang CF. Uniqueness theorems for differential pencils with eigenparameter boundary conditions and transmission conditions. J. Differ. Equ. 2013; 255 2615-2635.
[33] Yang CF, Yang XP. An interior inverse problem for the Sturm–Liouville operator with discontinuous conditions. Applied Mathematics Letters 2009; 22 (9): 1315-1319.
[34] Yang CF, Bondarenko NP, Xu XC. An inverse problem for the Sturm-Liouville pencil with arbitrary entire functions in the boundary condition. Inverse Problems & Imaging 2020; 14 (1): 153.
[35] Yurko VA. On determination of functional-differential pencils on closed sets from the Weyl-type function. Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform. 2020; 20 (3): 343–350.