Inverse problems for differential operators with nonseparated boundary conditions in the central symmetric case

Vjacheslav Yurko

Abstract


Inverse spectral problems for Sturm-Liouville operators on a finite interval with non-separated boundary conditions are studied in the central symmetric case, when the potential is symmetric with respect to the middle of the interval. We discuss statements of the problems, provide algorithms for their solutions along with necessary and sufficient conditions for the solvability of the inverse problems considered.

Keywords


differential operators; non-separated boundary conditions; inverse spectral problems

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References


V. A. Marchenko, Sturm-Liouville operators and their applications, Naukova Dumka, Kiev, 1977; English transl., Birkhauser, 1986.

B. M. Levitan, Inverse Sturm-Liouville problems. Nauka, Moscow, 1984; English transl., VNU Sci. Press, Utrecht, 1987.

J. Poschel and E. Trubowitz, Inverse Spectral Theory, Academic Press, New York, 1987.

G. Freiling and V. A. Yurko, Inverse Sturm-Liouville Problems and their Applications. NOVA Science Publishers, New York, 2001.

V. A. Yurko, Method of Spectral Mappings in the Inverse Problem Theory. Inverse and Ill-posed Problems Series. VSP, Utrecht, 2002.

I. V. Stankevich, An inverse problem of spectral analysis for Hill's equation, Doklady Akad. Nauk SSSR, 192, no.1(1970), 34--37. (in Russian); English transl. in Soviet Math. Dokl., 11(1970), 582--586.

V. A. Marchenko and I. V. Ostrovskii, A characterization of the spectrum of the Hill operator, Mathem. Sb., 97(1975), 540--606 (in Russian); English transl. in Math. USSR-Sb., 26(1975), 493--554.

V. A. Yurko, An inverse problem for second order differential operators with regular boundary conditions, Matem. Zametki, 18, no.4 (1975), 569--576 (in Russian); English transl. in Mathematical Notes, 18(1975), 928--932.

V. A. Yurko, On a periodic boundary value problem, Differ. Equations and Theory of Functions, Saratov Uni., Saratov, 1981, 109-115 (in Russian).

V. A. Yurko, On recovering differential operators with nonseparated boundary conditions, Study in Math. and Appl., Bashkir Uni., Ufa, 1981, 55-58 (in Russian).

O. A. Plaksina, Inverse problems of spectral analysis for the Sturm-Liouville operators with nonseparated boundary conditions, Mathem. Sb., 131(1986), 3--26 (in Russian); English transl. in Math. USSR-Sb.59(1988), 1--23.

I. M. Guseinov, M. G. Gasymov and I. M. Nabiev, An inverse problem for the Sturm-Liouville operator with nonseparable self-adjoint boundary conditions, Sib. Mathem. Zh., 31(1990), 46--54 (in Russian); English

trans. in Siberian Math. J., 31(1990), 910--918.

I. M. Guseinov and I. M. Nabiev, Solution of a class of inverse boundary-value Sturm-Liouville problems, Mathem. Sb., 186(1995),35--48 (in Russian); English transl. in Sbornik: Mathematics, 186(1995), 661--674.

P. Kargaev and E. Korotyaev, The inverse problem for the Hill operator, a direct approach, Invent. Math., 129(1997), 567-593.

V. A. Yurko, On differential operators with nonseparated boundary conditions, Funkt. Analiz i Prilozh., 28 (1994), 90--92 (Russian); English transl. in Functional Analysis and Applications, 28(1994), 295--297.

V. A. Yurko, The inverse spectral problem for differential operators with nonseparated boundary conditions, Journal of Mathematical Analysis and Applications, 250(2000), 266--289.

G. Freiling and V. A. Yurko, On the stability of constructing a potential in the central symmetry case, Applicable Analysis, 90(2011), 1819--1828.

Y. P. Wang, Inverse problems for discontinuos Sturm-Liouville operators with mixed spectral data, Inverse Problems in Sci. Eng.,23(2015), 1180--1198.

Y. P. Wang, C. F. Yang and Z. Y. Huang, Reconstruction for the spherically symmetric speed of sound from nodal data, Inverse Problems in Sci. Eng., 21(2013), 1032--1046.

C. T. Shieh and V. A. Yurko, Inverse spectral and nodal problems for discontinuous boundary value problems, Journal of Mathematical Analysis and Applications, 347(2008), 266--272.




DOI: http://dx.doi.org/10.5556/j.tkjm.48.2017.2492

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