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On discontinuous Dirac operator with eigenparameter dependent boundary and two transmission conditions. (English) Zbl 1342.34027
Summary: In this paper, we consider a discontinuous Dirac operator with eigenparameter dependent both boundary and two transmission conditions. We introduce a suitable Hilbert space formulation and get some properties of eigenvalues and eigenfunctions. Then we investigate the Green’s function, the resolvent operator, and some uniqueness theorems by using the Weyl function and some spectral data.

##### MSC:
 34A55 Inverse problems involving ordinary differential equations 34B24 Sturm-Liouville theory 34L05 General spectral theory of ordinary differential operators
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##### References:
 [1] Levitan, BM, Sargsyan, IS: Sturm-Liouville and Dirac Operators. Nauka, Moscow (1988) (in Russian) · Zbl 0657.34002 [2] Berezanskii, YM, Uniqueness theorem in the inverse spectral problem for the Schrödinger equation, Tr. Mosk. Mat. Obŝ., 7, 3-51, (1958) [3] Gasymov, MG; Dzhabiev, TT, Determination of a system of Dirac differential equations using two spectra, 46-71, (1975), Baku [4] Marchenko, VA: Sturm-Liouville Operators and Their Applications. Naukova Dumka, Kiev (1977) (in Russian) · Zbl 0399.34022 [5] Nizhnik, LP: Inverse Scattering Problems for Hyperbolic Equations. Naukova Dumka, Kiev (1977) (in Russian) · Zbl 0791.35142 [6] Gasymov, MG, Inverse problem of the scattering theory for Dirac system of order 2$$n$$, Tr. Mosk. Mat. Obŝ., 19, 41-112, (1968) · Zbl 0197.26102 [7] Guseinov, IM, On the representation of Jost solutions of a system of Dirac differential equations with discontinuous coefficients, Izv. Akad. Nauk Azerb. SSR, 5, 41-45, (1999) [8] Sat, M; Panakhov, ES, Spectral problem for diffusion operator, Appl. Anal., 93, 1178-1186, (2014) · Zbl 1296.34171 [9] Sat, M; Panakhov, ES, A uniqueness theorem for Bessel operator from interior spectral data, Abstr. Appl. Anal., 2013, (2013) · Zbl 1275.47097 [10] Fulton, CT, Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions, Proc. R. Soc. Edinb., 77A, 293-308, (1977) · Zbl 0376.34008 [11] Shkalikov, AA, Boundary value problems for ordinary differential equations with a parameter in boundary conditions, Tr. Semin. Im. I.G. Petrovskogo, 9, 190-229, (1983) · Zbl 0553.34014 [12] Yakubov, S: Completeness of Root Functions of Regular Differential Operators. Longman Scientific and Technical, New York (1994) · Zbl 0833.34081 [13] Kerimov, NB; Memedov, KK, On a boundary value problem with a spectral parameter in the boundary conditions, Sib. Mat. Zh., 40, 325-335, (1999) [14] Binding, PA; Browne, PJ; Watson, BA, Sturm-Liouville problems with boundary conditions rationally dependent on the eigenparameter II, J. Comput. Appl. Math., 148, 147-169, (2002) · Zbl 1019.34028 [15] Mukhtarov, OS; Kadakal, M; Muhtarov, FS, On discontinuous Sturm-Liouville problem with transmission conditions, J. Math. Kyoto Univ., 444, 779-798, (2004) · Zbl 1098.34069 [16] Tunç, E; Muhtarov, OS, Fundamental solution and eigenvalues of one boundary value problem with transmission conditions, Appl. Comput. Math., 157, 347-355, (2004) · Zbl 1060.34007 [17] Akdoğan, Z; Demirci, M; Mukhtarov, OS, Sturm-Liouville problems with eigendependent boundary and transmissions conditions, Acta Math. Sci., 25B, 731-740, (2005) · Zbl 1098.34068 [18] Akdoğan, Z; Demirci, M; Mukhtarov, OS, Discontinuous Sturm-Liouville problem with eigenparameter-dependent boundary and transmission conditions, Acta Appl. Math., 86, 329-344, (2005) · Zbl 1077.34029 [19] Fulton, CT, Singular eigenvalue problems with eigenvalue parameter contained in the boundary conditions, Proc. R. Soc. Edinb. A, 87, 1-34, (1980) · Zbl 0458.34013 [20] Amirov, RK; Ozkan, AS; Keskin, B, Inverse problems for impulsive Sturm-Liouville operator with spectral parameter linearly contained in boundary conditions, Integral Transforms Spec. Funct., 20, 607-618, (2009) · Zbl 1181.34019 [21] Guliyev, NJ, Inverse eigenvalue problems for Sturm-Liouville equations with spectral parameter linearly contained in one of the boundary conditions, Inverse Probl., 21, 1315-1330, (2005) · Zbl 1085.34004 [22] Mukhtarov, OS, Discontinuous boundary value problem with spectral parameter in boundary conditions, Turk. J. Math., 18, 183-192, (1994) · Zbl 0862.34016 [23] Tuna, H; Eryılmaz, A, Dissipative Sturm-Liouville operators with transmission conditions, Abstr. Appl. Anal., 2013, (2013) · Zbl 1291.34142 [24] Russakovskii, EM, Operator treatment of boundary problems with spectral parameters entering via polynomials in the boundary conditions, Funct. Anal. Appl., 9, 358-359, (1975) · Zbl 0339.34022 [25] Binding, PA; Browne, PJ; Seddighi, K, Sturm-Liouville problems with eigenparameter dependent boundary conditions, Proc. Edinb. Math. Soc., 37, 57-72, (1993) · Zbl 0791.34023 [26] Russakovskii, EM, Matrix boundary value problems with eigenvalue dependent boundary conditions, No. 95, 453-462, (1997), Basel · Zbl 0922.34011 [27] Mennicken, R; Schmid, H; Shkalikov, AA, On the eigenvalue accumulation of Sturm-Liouville problems depending nonlinearly on the spectral parameter, Math. Nachr., 189, 157-170, (1998) · Zbl 0892.34019 [28] Binding, PA; Browne, PJ; Watson, BA, Inverse spectral problems for Sturm-Liouville equations with eigenparameter dependent boundary conditions, J. Lond. Math. Soc., 62, 161-182, (2000) · Zbl 0960.34010 [29] Schmid, H; Tretter, C, Singular Dirac systems and Sturm-Liouville problems nonlinear in the spectral parameter, J. Differ. Equ., 181, 511-542, (2002) · Zbl 1012.34082 [30] Binding, PA; Browne, PJ; Watson, BA, Equivalence of inverse Sturm-Liouville problems with boundary conditions rationally dependent on the eigenparameter, J. Math. Anal. Appl., 29, 246-261, (2004) · Zbl 1046.34021 [31] Hald, OH, Discontinuous inverse eigenvalue problems, Commun. Pure Appl. Math., 37, 539-577, (1984) · Zbl 0541.34012 [32] Kobayashi, M, A uniqueness proof for discontinuous inverse Sturm-Liouville problems with symmetric potentials, Inverse Probl., 5, 767-781, (1989) · Zbl 0698.34021 [33] Shepelsky, D, The inverse problem of reconstruction of the medium’s conductivity in a class of discontinuous and increasing functions, No. 19, 209-232, (1994), Providence · Zbl 0823.34019 [34] Amirov, RK; Güldü, Y, Inverse problems for Dirac operator with discontinuity conditions inside an interval, Int. J. Pure Appl. Math., 37, 215-226, (2007) · Zbl 1142.34005 [35] Likov, AV, Mikhailov, YA: The Theory of Heat and Mass Transfer. Qosenergaizdat, Moscow (1963) (in Russian) [36] Meschanov, VP, Feldstein, AL: Automatic Design of Directional Couplers. Sviaz, Moscow (1980) [37] Tikhonov, AN, Samarskii, AA: Equations of Mathematical Physics. Pergamon, Oxford (1990) · Zbl 0044.09302 [38] McLaughlin, J; Polyakov, P, On the uniqueness of a spherical symmetric speed of sound from transmission eigenvalues, J. Differ. Equ., 107, 351-382, (1994) · Zbl 0803.35163 [39] Voitovich, NN, Katsenelbaum, BZ, Sivov, AN: Generalized Method of Eigen-Vibration in the Theory of Diffraction. Nauka, Moscow (1997) (in Russian) [40] Titeux, I; Yakubov, Y, Completeness of root functions for thermal conduction in a strip with peicewise continuous coefficients, Math. Models Methods Appl. Sci., 7, 1035-1050, (1997) · Zbl 0903.73008 [41] Yurko, VA, Integral transforms connected with discontinuous boundary value problems, Integral Transforms Spec. Funct., 10, 141-164, (2000) · Zbl 0989.34015 [42] Freiling, G, Yurko, VA: Inverse Sturm-Liouville Problems and Their Applications. Nova Science, New York (2001) · Zbl 1037.34005 [43] Kadakal, M; Mukhtarov, OS, Sturm-Liouville problems with discontinuities at two points, Comput. Math. Appl., 54, 1367-1379, (2007) · Zbl 1140.34012 [44] Yang, Q; Wang, W, Asymptotic behavior of a differential operator with discontinuities at two points, Math. Methods Appl. Sci., 34, 373-383, (2011) · Zbl 1220.34043 [45] Shahriari, M; Akbarfam, AJ; Teschl, G, Uniqueness for inverse Sturm-Liouville problems with a finite number of transmission conditions, J. Math. Anal. Appl., 395, 19-29, (2012) · Zbl 1251.34025 [46] Güldü, Y, Inverse eigenvalue problems for a discontinuous Sturm-Liouville operator with two discontinuities, Bound. Value Probl., 2013, (2013) · Zbl 1297.34021 [47] Yang, C-F, Uniqueness theorems for differential pencils with eigenparameter boundary conditions and transmission conditions, J. Differ. Equ., 255, 2615-2635, (2013) · Zbl 1288.34016 [48] Zhdanovich, VF, Formulae for the zeros of Dirichlet polynomials and quasi-polynomials, Dokl. Akad. Nauk SSSR, 135, 1046-1049, (1960) · Zbl 0111.27101 [49] Titchmarsh, EC: The Theory of Functions. Oxford University Press, London (1939) · Zbl 0022.14602
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