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Inverse problem for interior spectral data of the Dirac operator on a finite interval. (English) Zbl 1021.34073
The authors consider the Dirac operator, generated by the differential expression $$ l(y)=By'+Qy, \quad 0\leq x\leq 1, $$ with $$ B=\pmatrix 0 & 1 \cr -1 & 0 \endpmatrix, \quad Q(x)=\pmatrix p(x) & q(x) \cr q(x) & -p(x)\endpmatrix, \quad y(x)=\pmatrix y_1(x) \cr y_2(x)\endpmatrix,$$ subject to the boundary conditions $$ y_1(0)\cos\alpha+y_2(0)\sin\alpha=0,\quad \alpha \in[0,\pi),\quad y_1(1)\cos\beta+y_2(1)\sin\beta=0,\quad \beta\in[0,\pi).$$ The authors prove uniqueness theorems taking the spectrum and the set of values of the eigenfunctions at some interior point as the given data.

34L40Particular ordinary differential operators
34B24Sturm-Liouville theory
34L05General spectral theory for OD operators
Full Text: DOI
[1] Arutyunyan, T. N., Isospectral Dirac operators, Izv. Nats. Akad. Nauk Armenii Mat., 29 (1994), no. 2, 3-14; English transl. in J. Contemp. Math. Anal., Armen. Acad. Sci., 29 (1994), no. 2, 1-10. · Zbl 0830.35106
[2] Borg, G., Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe, Bestimmung der Differential gleichung durch die Eigenvert, Acta Math., 78 (1946), 1-96. · Zbl 0063.00523 · doi:10.1007/BF02421600
[3] Gasymov, M. G. and Levitan, B. M., The Inverse Problem for a Dirac System, Dokl. Akad. Nauk SSSR, 167 (1966), 967-970.
[4] Hochstadt, H. and Lieberman, B., An Inverse Sturm-Liouville Problem with Mixed Given Data, SIAM J. Appl. Math., 34 (1978), 676-680. · Zbl 0418.34032 · doi:10.1137/0134054
[5] Levin, B. Ja., Distribution of zeros of entire functions, AMS Transl. Vol.5, Providence, 1964. · Zbl 0152.06703
[6] Levitan, B. M. and Sargsjan, I. S., Sturm-Liouville and Dirac Operators, Kluwer Aca- demic Publishers, Dodrecht, Boston, London, 1991.
[7] Malamud, M. M., Uniqueness Questions in Inverse Problems for Systems of Differential Equations on a Finite Interval, Trans. Moscow Math. Soc., 60 (1999), 173-224.
[8] Marchenko, V. A., On certain questions in the theory of differential operators of the second order, Dokl. Akad. Nauk SSSR, 72 (1950), 457-460 (in Russian).
[9] ---, Some questions in the theory of one-dimensional linear differential operators of the second order. I, Trudy Moscov. Mat. Obsc., 1 (1952), 327-420 (in Russian); Amer. Math. Soc. Transl., Ser. 2 101 (1973), 1-104.
[10] Mochizuki, K. and Trooshin, I., Inverse Problem for Interior Spectral Data of Sturm- Liouville Operator, Seminar Notes of Mathematical Sciences, 3 Ibaraki University, (2000), 44-51. · Zbl 1035.34008 · doi:10.1515/jiip.2001.9.4.425
[11] ---, Inverse Problem for Interior Spectral Data of Sturm-Liouville Operator, J. In- verse Ill-posed Prob., 9 (2001), 425-433. · Zbl 1035.34008 · doi:10.1515/jiip.2001.9.4.425
[12] Ramm, A. G., Property C for ODE and applications to inverse problems, in Opera- tor theory and applications (Winnipeg, MB, 1998), Fields Inst. Commun., 25, AMS, Providence, RI, (2000), 15-75. i i i i · Zbl 0964.34078