journal article
Open Access
Feb 24, 2023
Inverse Sturm–Liouville Problem with Spectral Parameter in the Boundary Conditions
Abstract
In this paper, for the first time, we study the inverse Sturm–Liouville problem with polynomials of the spectral parameter in the first boundary condition and with entire analytic functions in the second one. For the investigation of this new inverse problem, we develop an approach based on the construction of a special vector functional sequence in a suitable Hilbert space. The uniqueness of recovering the potential and the polynomials of the boundary condition from a part of the spectrum is proved. Furthermore, our main results are applied to the Hochstadt–Lieberman-type problems with polynomial dependence on the spectral parameter not only in the boundary conditions but also in discontinuity (transmission) conditions inside the interval. We prove novel uniqueness theorems, which generalize and improve the previous results in this direction. Note that all the spectral problems in this paper are investigated in the general non-self-adjoint form, and our method does not require the simplicity of the spectrum. Moreover, our method is constructive and can be developed in the future for numerical solution and for the study of solvability and stability of inverse spectral problems.
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References
48
[1]
Marchenko, V.A. (1986). Sturm–Liouville Operators and Their Applications, Birkhäuser.
10.1007/978-3-0348-5485-6
[2]
Levitan, B.M. (1987). Inverse Sturm–Liouville Problems, VNU Sci. Press.
10.1515/9783110941937
[3]
Pöschel, J., and Trubowitz, E. (1987). Inverse Spectral Theory, Academic Press.
[4]
Freiling, G., and Yurko, V. (2001). Inverse Sturm–Liouville Problems and Their Applications, Nova Science Publishers.
[5]
Collatz, L. (1963). Eigenwertaufgaben mit technischen Anwendungen, Akad, Verlagsgesellschaft Geest & Portig.
[6]
Kraft "Adjointness properties for differential systems with eigenvalue-dependent boundary conditions, with application to flow-duct acoustics" J. Acoust. Soc. Am. (1977) 10.1121/1.381383
[7]
Fulton "Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions" Proc. R. Soc. Edinb. Sect. A (1977) 10.1017/s030821050002521x
[8]
Fulton "Singular eigenvalue problems with eigenvalue parameter contained in the boundary conditions" Proc. R. Soc. Edinb. Sect. A (1980) 10.1017/s0308210500012312
[9]
Mennicken, R., and Möller, M. (2003). Non-Self-Adjoint Boundary Eigenvalue Problems, Elsevier.
[10]
Shkalikov "Boundary problems for ordinary problems for differential Equations with parameter in the boundary conditions" J. Sov. Math. (1986) 10.1007/bf01084754
[11]
Tretter "Boundary eigenvalue problems with differential Equations Nη = λPη with λ-polynomial boundary conditions" J. Diff. Eqns. (2001) 10.1006/jdeq.2000.3829
[12]
Chugunova "Inverse spectral problem for the Sturm–Liouville operator with eigenvalue parameter dependent boundary conditions" Oper. Theory Advan. Appl. (2001)
[13]
Binding "Sturm–Liouville problems with boundary conditions rationally dependent on the eigenparameter. I" Proc. Edinb. Math. Soc. (2002) 10.1017/s0013091501000773
[14]
Binding "Sturm–Liouville problems with boundary conditions rationally dependent on the eigenparameter. II" J. Comput. Appl. Math. (2002) 10.1016/s0377-0427(02)00579-4
[15]
Binding "Equivalence of inverse Sturm–Liouville problems with boundary conditions rationally dependent on the eigenparameter" J. Math. Anal. Appl. (2004) 10.1016/j.jmaa.2003.11.025
[16]
Chernozhukova "A uniqueness theorem for the boundary value problems with non-linear dependence on the spectral parameter in the boundary conditions" Inv. Probl. Sci. Eng. (2009) 10.1080/17415970802538550
[17]
Freiling "Inverse problems for Sturm–Liouville Equations with boundary conditions polynomially dependent on the spectral parameter" Inverse Probl. (2010) 10.1088/0266-5611/26/5/055003
[18]
Freiling "Determination of singular differential pencils from the Weyl function" Adv. Dynam. Syst. Appl. (2012)
[19]
Wang "Uniqueness theorems for Sturm–Liouville operators with boundary conditions polynomially dependent on the eigenparameter from spectral data" Results Math. (2013) 10.1007/s00025-012-0258-6
[20]
Yang "Ambarzumyan-type theorem with polynomially dependent eigenparameter" Math. Meth. Appl. Sci. (2015) 10.1002/mma.3380
[21]
Yang "Inverse scattering problems for Sturm–Liouville operators with spectral parameter dependent on boundary conditions" Math. Notes (2018) 10.1134/s0001434618010078
[22]
Mosazadeh "On Hochstadt-Lieberman theorem for impulsive Sturm–Liouville problems with boundary conditions polynomially dependent on the spectral parameter" Turk. J. Math. (2018) 10.3906/mat-1807-77
[23]
Guliyev "Schrödinger operators with distributional potentials and boundary conditions dependent on the eigenvalue parameter" J. Math. Phys. (2019) 10.1063/1.5048692
[24]
Guliyev "Essentially isospectral transformations and their applications" Ann. Di Mat. Pura Ed Appl. (2020) 10.1007/s10231-019-00934-w
[25]
Guliyev "On two-spectra inverse problems" Proc. AMS (2020) 10.1090/proc/15155
[26]
Browne "A uniqueness theorem for inverse eigenparameter dependent Sturm–Liouville problems" Inverse Probl. (1997) 10.1088/0266-5611/13/6/003
[27]
Guliyev "Inverse eigenvalue problems for Sturm–Liouville Equations with spectral parameter linearly contained in one of the boundary condition" Inverse Probl. (2005) 10.1088/0266-5611/21/4/008
[28]
Buterin "On half inverse problem for differential pencils with the spectral parameter in boundary conditions" Tamkang J. Math. (2011) 10.5556/j.tkjm.42.2011.912
[29]
Guliyev, N.J. (2001). Inverse square singularities and eigenparameter dependent boundary conditions are two sides of the same coin. arXiv.
[30]
Bondarenko "Inverse Sturm–Liouville problem with analytical functions in the boundary condition" Open Math. (2020) 10.1515/math-2020-0188
[31]
Bondarenko "Solvability and stability of the inverse Sturm–Liouville problem with analytical functions in the boundary condition" Math. Meth. Appl. Sci. (2020) 10.1002/mma.6451
[32]
Yang "An inverse problem for the Sturm–Liouville pencil with arbitrary entire functions in the boundary condition" Inverse Probl. Imag. (2020) 10.3934/ipi.2019068
[33]
Bondarenko "A partial inverse Sturm–Liouville problem on an arbitrary graph" Math. Meth. Appl. Sci. (2021) 10.1002/mma.7231
[34]
Kuznetsova, M.A. (2022). On recovering quadratic pencils with singular coefficients and entire functions in the boundary conditions. Math. Meth. Appl. Sci.
10.1002/mma.8819
[35]
Hochstadt "An inverse Sturm–Liouville problem with mixed given data" SIAM J. Appl. Math. (1978) 10.1137/0134054
[36]
Borg "Eine Umkehrung der Sturm–Liouvilleschen Eigenwertaufgabe: Bestimmung der Differentialgleichung durch die Eigenwerte" Acta Math. (1946) 10.1007/bf02421600
[37]
Bondarenko "A partial inverse problem for the Sturm–Liouville operator on a star-shaped graph" Anal. Math. Phys. (2018) 10.1007/s13324-017-0172-x
[38]
Bondarenko "On a local solvability and stability of the inverse transmission eigenvalue problem" Inverse Probl. (2017) 10.1088/1361-6420/aa8cb5
[39]
Bondarenko "A new approach to the inverse discrete transmission eigenvalue problem" Inverse Probl. Imag. (2022) 10.3934/ipi.2021073
[40]
"Inverse spectral problems and closed exponential systems" Ann. Math. (2005) 10.4007/annals.2005.162.885
[41]
Kravchenko "A practical method for recovering Sturm–Liouville problems from the Weyl function" Inverse Probl. (2021) 10.1088/1361-6420/abff06
[42]
Bartels "Sturm–Liouville problems with transfer condition Herglotz dependent on the eigenparameter: Hilbert space formulation" Integr. Equ. Oper. Theory (2018) 10.1007/s00020-018-2463-5
[43]
Bartels "Sturm–Liouville Problems with transfer condition Herglotz dependent on the eigenparameter: Eigenvalue asymptotics" Complex Anal. Oper. Theory (2021) 10.1007/s11785-021-01119-1
[44]
Ozkan "Spectral problems for Sturm–Liouville operator with boundary and jump conditions linearly dependent on the eigenparameter" Inv. Probl. Sci. Engin. (2012) 10.1080/17415977.2011.652957
[45]
Wei "Inverse spectral problem for non selfadjoint Dirac operator with boundary and jump conditions dependent on the spectral parameter" J. Comput. Appl. Math. (2016) 10.1016/j.cam.2016.05.018
[46]
Kuznetsova "A uniqueness theorem on inverse spectral problems for the Sturm–Liouville differential operators on time scales" Results Math. (2020) 10.1007/s00025-020-1171-z
[47]
Kuznetsova "On recovering the Sturm–Liouville differential operators on time scales" Math. Notes (2021) 10.1134/s0001434621010090
[48]
Da Silva, S.L.E. (2020). F; Carvalho, P.T.C.; da Costa, C.A.N; de Araujo, J.M; Corso, G. An objective function for full-waveform inversion based on -dependent offset-preconditioning. PLoS ONE, 15.
10.1371/journal.pone.0240999
Metrics
17
Citations
48
References
Details
- Published
- Feb 24, 2023
- Vol/Issue
- 11(5)
- Pages
- 1138
- License
- View
Authors
Funding
Russian Science Foundation
Award: 21-71-10001
Cite This Article
Natalia P. Bondarenko, Egor E. Chitorkin (2023). Inverse Sturm–Liouville Problem with Spectral Parameter in the Boundary Conditions. Mathematics, 11(5), 1138. https://doi.org/10.3390/math11051138
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