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An interior inverse problem for the Sturm-Liouville operator with discontinuous conditions. (English) Zbl 1173.34306
Summary: We study the inverse problem for the Sturm-Liouville operator $-D^{2}+q$ with discontinuity boundary conditions inside a finite closed interval. Using spectral data of a kind, it is shown that the potential function $q(x)$ can be uniquely determined by a set of values of eigenfunctions at some internal point and one spectrum.

34A55Inverse problems of ODE
34L20Asymptotic distribution of eigenvalues for OD operators
47E05Ordinary differential operators
Full Text: DOI
[1] Alpay, D.; Gohberg, I.: Inverse problems associated to a canonical differential system, Oper. theory adv. Appl. 127, 1-27 (2001) · Zbl 0991.34070
[2] Gelfand, I. M.; Levitan, B. M.: On the determination of a differential equation from its spectral function, Izv. akad. Nauk SSR. Ser. mat. 15, 309-360 (1951)
[3] Levitan, B. M.: On the determination of the Sturm--Liouville operator from one and two spectra, Math. USSR izv. 12, 179-193 (1978) · Zbl 0401.34022 · doi:10.1070/IM1978v012n01ABEH001844
[4] Levitan, B. M.: Inverse Sturm--Liouville problems, (1987) · Zbl 0749.34001
[5] Ambarzumyan, V. A.: Über eine frage der eigenwerttheorie, Z. phys. 53, 690-695 (1929) · Zbl 55.0868.01
[6] Anderssen, R. S.: The effect of discontinuities in density and shear velocity on the asymptotic overtone structure of tortional eigenfrequencies of the Earth, Geophys. J. R. astron. Soc. 50, 303-309 (1997)
[7] Hald, O. H.: Discontinuous inverse eigenvalue problem, Commun. pure appl. Math. 37, 539-577 (1984) · Zbl 0541.34012 · doi:10.1002/cpa.3160370502
[8] Krueger, R. J.: Inverse problems for nonabsorbing media with discontinuous material properties, J. math. Phys. 23, 396-404 (1982) · Zbl 0511.35079 · doi:10.1063/1.525358
[9] Freiling, G.; Yurko, V. A.: Inverse spectral problems for singular non-selfadjoint differential operators with discontinuities in an interior point, Inverse problems 18, 757-773 (2002) · Zbl 1012.34083 · doi:10.1088/0266-5611/18/3/316
[10] Yurko, V. A.: Integral transforms connected with discontinuous boundary value problems, Integral transforms spec. Funct. 10, No. 2, 141-164 (2000) · Zbl 0989.34015 · doi:10.1080/10652460008819282
[11] Hochstadt, H.; Lieberman, B.: An inverse Sturm--Liouville problem with mixed given data, SIAM J. Appl. math. 34, 676-680 (1978) · Zbl 0418.34032 · doi:10.1137/0134054
[12] Ramm, A. G.: Property C for ODE and applications to inverse problems, Fields inst. Commun. 25, 15-75 (2000) · Zbl 0964.34078
[13] Mochizuki, K.; Trooshin, I.: Inverse problem for interior spectral data of Sturm--Liouville operator, J. inverse ill-posed probl. 9, 425-433 (2001) · Zbl 1035.34008
[14] Mochizuki, K.; Trooshin, I.: Inverse problem for interior spectral data of the Dirac operator on a finite interval, Publ. RIMS, Kyoto univ. 38, 387-395 (2002) · Zbl 1021.34073 · doi:10.2977/prims/1145476343
[15] Shieh, C. T.; Yurko, V. A.: Inverse nodal and inverse spectral problems for discontinuous boundary value problems, J. math. Anal. appl. 347, 266-272 (2008) · Zbl 1209.34014 · doi:10.1016/j.jmaa.2008.05.097
[16] Yurko, V. A.: Method of spectral mappings in the inverse problem theory, Inverse ill-posed probl. Ser. (2002) · Zbl 1098.34008
[17] Freiling, G.; Yurko, V. A.: Inverse Sturm--Liouville problems and their applications, (2001) · Zbl 1037.34005
[18] Marchenko, V. A.: Sturm--Liouville operators and their applications, (1977) · Zbl 0399.34022
[19] Borg, G.: Eine umkehrung der Sturm--liouvillesehen eigenwertaufgabe, Acta math. 78, 1-96 (1946) · Zbl 0063.00523
[20] Carlson, R.: Inverse spectral theory for some singular Sturm--Liouville problems, J. differential equations 106, No. 1, 121-140 (1993) · Zbl 0813.34015 · doi:10.1006/jdeq.1993.1102
[21] Cheng, Y. H.; Law, C. K.; Tsay, J.: Remarks on a new inverse nodal problem, J. math. Anal. appl. 248, 145-155 (2000) · Zbl 0960.34018 · doi:10.1006/jmaa.2000.6878
[22] Gesztesy, F.; Simon, B.: Inverse spectral analysis with partial information on the potential II: The case of discrete spectrum, Trans. amer. Math. soc. 352, No. 6, 2765-2787 (2000) · Zbl 0948.34060 · doi:10.1090/S0002-9947-99-02544-1
[23] Hald, O. H.; Mclaughlin, J. R.: Solutions of inverse nodal problems, Inverse problems 5, 307-347 (1989) · Zbl 0667.34020 · doi:10.1088/0266-5611/5/3/008
[24] Horvath, M.: Inverse spectral problems and closed exponential systems, Ann. of math. 162, 885-918 (2005) · Zbl 1102.34005 · doi:10.4007/annals.2005.162.885
[25] Law, C. K.; Yang, C. F.: Reconstructing the potential function and its derivatives using nodal data, Inverse problems 14, 299-312 (1998) · Zbl 0901.34023 · doi:10.1088/0266-5611/14/2/006
[26] Levitan, B. M.; Sargsjan, I. S.: Sturm--Liouville and Dirac operators, (1991) · Zbl 0106.06004
[27] Marchenko, V. A.: On certain questions in the theory of differential operators of the second order, Dokl. akad. Nauk SSSR 72, 457-460 (1950)
[28] Marchenko, V. A.: Some questions in the theory of one-dimensional linear differential operators of the second order, I, Trudy moscov. Mat. obsc. 1, 327-420 (1952)
[29] Mclaughlin, J. R.: Inverse spectral theory using nodal points as data--a uniqueness result, J. differential equations 73, 354-362 (1988) · Zbl 0652.34029 · doi:10.1016/0022-0396(88)90111-8
[30] Pöschel, J.; Trubowitz, E.: Inverse spectral theory, (1987)
[31] Yang, X. F.: A solution of the inverse nodal problem, Inverse problems 13, 203-213 (1997) · Zbl 0873.34017 · doi:10.1088/0266-5611/13/1/016