Structure of the spectrum of a nonselfadjoint Dirac operator
Sbornik. Mathematics, Tome 214 (2023) no. 1, pp. 39-57 Cet article a éte moissonné depuis la source Math-Net.Ru

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For the Dirac operator with two-point boundary conditions and an arbitrary complex-valued $L_2$-integrable potential $V(x)$ the spectral problem is considered. Necessary and sufficient conditions on an entire function to be the characteristic function of such a boundary value problem are obtained. Necessary and sufficient conditions on the spectrum of the above operator are established in the case when the boundary conditions are regular. Bibliography: 16 titles.
Keywords: Dirac operator, characteristic function, spectrum.
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A. S. Makin. Structure of the spectrum of a nonselfadjoint Dirac operator. Sbornik. Mathematics, Tome 214 (2023) no. 1, pp. 39-57. http://geodesic.mathdoc.fr/item/SM_2023_214_1_a2/

[1] M. G. Gasymov and T. T. Dzhabiev, “Solution of the inverse problem by two spectra for the Dirac equation on a finite interval”, Dokl. Akad. Nauk Azerb. SSR, 22:7 (1966), 3–7 (Russian) | MR | Zbl

[2] S. Albeverio, R. Hryniv and Y. Mykytyuk, “Inverse spectral problems for Dirac operators with summable potentials”, Russ. J. Math. Phys., 12:4 (2005), 406–423 | MR | Zbl

[3] T. V. Misyura, “Characteristics of the spectra of the periodic and antiperiodic boundary value problems that are generated by the Dirac operator. II”, Teor. Funktsii Funktsional. Anal. i Prilozhen., 31, Khar'kov University Publishing House, Khar'kov, 1979, 102–109 (Russian) | MR | Zbl

[4] I. M. Nabiev, “Solution of the inverse quasiperiodic problem for the Dirac system”, Mat. Zametki, 89:6 (2011), 885–893 ; English transl. in Math. Notes, 89:6 (2011), 845–852 | DOI | MR | Zbl | DOI

[5] V. A. Yurko, “An inverse spectral problem for singular non-self-adjoint differential systems”, Mat. Sb., 195:12 (2004), 123–156 ; English transl. in Sb. Math., 195:12 (2004), 1823–1854 | DOI | MR | Zbl | DOI

[6] V. Yurko, “Inverse spectral problems for differential systems on a finite interval”, Results Math., 48:3–4 (2005), 371–386 | DOI | MR | Zbl

[7] A. S. Makin, “On the spectrum of two-point boundary value problems for the Dirac operator”, Differ. Uravn., 57:8 (2021), 1023–1031 ; English transl. in Differ. Equ., 57:8 (2021), 993–1002 | MR | Zbl | DOI

[8] V. Tkachenko, “Non-self-adjoint periodic Dirac operators”, Operator theory, system theory and related topics (Beer-Sheva/Rehovot 1997), Oper. Theory Adv. Appl., 123, Birkhäuser, Basel, 2001, 485–512 | MR | Zbl

[9] V. A. Marchenko, Sturm-Liouville operators and applications, Naukova Dumka, Kiev, 1977, 331 pp. ; English transl., Oper. Theory Adv. Appl., 22, Birkhäuser Verlag, Basel, 1986, xii+367 pp. | MR | Zbl | DOI | MR | Zbl

[10] V. Tkachenko, “Non-selfadjoint periodic Dirac operators with finite-band spectra”, Integral Equations Operator Theory, 36:3 (2000), 325–348 | DOI | MR | Zbl

[11] B. Ya. Levin, Lectures on entire functions, Moscow State University, Moscow, 1971, 124 pp.; English transl., Transl. Math. Monogr., 150, Amer. Math. Soc., Providence, RI, 1996, xvi+248 pp. | MR | Zbl

[12] S. M. Nikol'skiĭ, Approximation of functions of several variables and imbedding theorems, 2nd ed., Nauka, Moscow, 1977, 455 pp. ; English transl. of 1st ed., Grundlehren Math. Wiss., 205, Springer-Verlag, New York–Heidelberg, 1975, viii+418 pp. | MR | Zbl | DOI | MR | Zbl

[13] B. J. Levin and I. V. Ostrovskii, “On small perturbations of the set of zeros of functions of sine type”, Izv. Akad. Nauk SSSR Ser. Mat., 43:1 (1979), 87–110 ; English transl. in Math. USSR-Izv., 14:1 (1980), 79–101 | MR | Zbl | DOI

[14] P. Djakov and B. Mityagin, “Unconditional convergence of spectral decompositions of 1D Dirac operators with regular boundary conditions”, Indiana Univ. Math. J., 61:1 (2012), 359–398 | DOI | MR | Zbl

[15] M. A. Lavrentiev and B. V. Shabat, Methods of the theory of functions of complex variable, 4th ed., Nauka, Moscow, 1973, 736 pp. ; German transl. of 3d ed., M. A. Lawrentjew and B. W. Schabat, Methoden der komplexen Funktionentheorie, Math. Naturwiss. Tech., 13, VEB Deutscher Verlag der Wissenschaften, Berlin, 1967, x+846 pp. | Zbl | MR | Zbl

[16] J.-J. Sansuc and V. Tkachenko, “Characterization of the periodic and anti-periodic spectra of nonselfadjoint Hill's operators”, New results in operator theory and its applications, Oper. Theory Adv. Appl., 98, Birkhäuser, Basel, 1997, 216–224 | DOI | MR | Zbl