Hostname: page-component-848d4c4894-hfldf Total loading time: 0 Render date: 2024-06-12T01:48:23.041Z Has data issue: false hasContentIssue false
  • English
  • Français

On the spectrum of non-self-adjoint Dirac operators with quasi-periodic boundary conditions

Part of: General theory in ordinary differential equations Ordinary differential operators

Published online by Cambridge University Press:  26 May 2022

Alexander Makin Open the ORCID record for Alexander Makin [Opens in a new window]
Alexander Makin*
Affiliation:
Russian Technological University, Prospect Vernadskogo 78, Moscow 119454, Russia (alexmakin@yandex.ru)
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we consider non-self-adjoint Dirac operators on a finite interval with complex-valued potentials and quasi-periodic boundary conditions. Necessary and sufficient conditions for a set of complex numbers to be the spectrum of the indicated problem are established.

Keywords

Dirac operatorinverse problemquasi-periodic boundary conditionsspectrum

MSC classification

Primary: 34L40: Particular operators (Dirac, one-dimensional Schrödinger, etc.)
Secondary: 34A55: Inverse problems
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 153 , Issue 4 , August 2023 , pp. 1099 - 1117
Check for updates
Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Albeverio, A., Hryniv, R. and Mykytyuk, Ya.. Inverse spectral problems for Dirac operators with summable potentials. Russ. J. Math. Phys. 12 (2005), 406–423.Google Scholar
Birkhoff, G. D. and Langer, R. E.. The boundary problems and developments associated with a system of ordinary differential equations of the first order. Proc. Am. Acad. Arts Sci. 58 (1923), 49–128.CrossRefGoogle Scholar
Bondarenko, N. and Buterin, S.. An inverse spectral problem for integro-differential Dirac operators with general convolution kernels. Appl. Anal. 99 (2020), 700–716.Google Scholar
Daskalov, Vasil B. and Khristov, Evgeni Kh.. Explicit formulae for the inverse problem for the regular Dirac operator. Inverse Probl. 16 (2000), 247–258.Google Scholar
Djakov, P. and Mityagin, B.. Unconditional convergence of spectral decompositions of 1D Dirac operators with regular boundary conditions. Indiana Univ. Math. J. 61 (2012), 359–398.CrossRefGoogle Scholar
Dzabiev, T. T.. The inverse problem for the Dirac equation with a singularity. Acad. Nauk. Azerbaidzan. SSR. Dokl. 22 (1966), 8–12 (in Russian).Google Scholar
Gasymov, M. G. and Dzabiev, T. T.. Solution of the inverse problem by two spectra for the Dirac equation on a finite interval. Acad. Nauk. Azerbaidzan. SSR. Dokl. 22 (1966), 3–7 (in Russian).Google Scholar
Gesztesy, F. and Sakhnovich, A.. The inverse approach to Dirac-type systems based on the A-function concept. J. Funct. Anal. 279 (2020), 108609.CrossRefGoogle Scholar
Gorbunov, O. and Yurko, V.. Inverse problem for Dirac system with singularities in interior points. Anal Math. Phys. 6 (2016), 1–29.CrossRefGoogle Scholar
Lavrentiev, M. A. and Shabat, B. V.. Methods of Theory of Complex Variable (Nauka, Moscow, 1973) (in Russian).Google Scholar
Lesch, M. and Malamud, M., The inverse spectral problem for first order systems on the half line, Differential operators and related topics. Proceedings of the Mark Krein international conference on operator theory and applications, Odessa, Ukraine, August 18–22, 1997. Volume I (Basel, Birkhauser, 2000). Oper. Theory, Adv. Appl. 117, 199–238.Google Scholar
Levin, B. Ya., Lectures on Entire Functions, Am. Math. Soc. Transl. Math. Monographs Vol. 150 (Am. Math. Providence, RI, 1996).Google Scholar
Levin, B. Ya. and Ostrovskii, I. V.. On small perturbations of the set of zeros of functions of sine type. Math. USSR-Izv. 14 (1980), 79–101.Google Scholar
Levitan, B. M. and Sargsyan, I. S.. Sturm–Liouville and Dirac operators (Kluwer Academic Publishers, Dordrecht, 1991).CrossRefGoogle Scholar
Malamud, M. M.. On Borg-type theorems for first-order systems on a finite interval. Funktsional. Anal. i Prilozhen. 33 (1999), 75–80 (in Russian); Engl. transl.: Funct. Anal. Appl. 33 (1999), no. 1, 64–68.Google Scholar
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.Google Scholar
Malamud, M. M.. Unique determination of a system by a part of the Monodromy matrix. Func. Anal. Appl. 49 (2015), 264–278.CrossRefGoogle Scholar
Mamedov, S. G.. The inverse boundary problem on a finite interval for the Dirac's systems of equations. Azerbaidzan Gos. Univ. Uchen. Zap. Ser. Fiz-Mat. Nauk. 57 (1975), 61–67 (in Russian).Google Scholar
Marchenko, V. A.. Sturm–Liouville operators and their applications (Birkhauser Verlag, Basel, 1986).CrossRefGoogle Scholar
Misyura, T. V.. Characterization of spectra of periodic and anti-periodic problems generated by Dirac's operators. II. Theoriya functfii, funct. analiz i ikh prilozhen. 31 (1979), 102–109.Google Scholar
Mykytyuk, Y. V. and Puyda, D. V.. Inverse spectral problems for Dirac operators on a finite interval. J. Math. Anal. Appl. 386 (2012), 177–194.CrossRefGoogle Scholar
Nabiev, I. M.. Solution of the class of inverse problems for the Dirac operators. Trans. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. Math. Mech. 21 (2001), 146–157.Google Scholar
Nabiev, I. M.. Characteristic of spectral data of Dirac operators. Trans. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. Math. Mech. 24 (2004), 161–166.Google Scholar
Nabiev, I. M.. Solution of the quasiperiodic problem for the Dirac system. Math. Notes 89 (2011), 845–852.CrossRefGoogle Scholar
Nabiev, I. M.. The inverse periodic problem for the Dirac operator. Proc. IMM NAS Azerbaijan XIX (2003), 177–180.Google Scholar
Nikolskii, S. M.. Approximation of Functions of Several Variables and Embedding Theorems (Nauka, Moscow, 1977) (in Russian).Google Scholar
Ning, W.. An inverse spectral problem for a nonsymmetric differential operator: Reconstruction of eigenvalue problem. J. Math. Anal. Appl. 327 (2007), 1396–1419.CrossRefGoogle Scholar
Sansug, J.-J. and Tkachenko, V.. Characterization of the periodic and antiperiodic spectra of nonselfadjoint Hill's operators. Oper. Theory Adv. Appl. 98 (1997), 216–224.Google Scholar
Tkachenko, V.. Non-self-adjoint periodic Dirac operators. Oper. Theory: Adv. Appl. 123 (2001), 485–512.Google Scholar
Tkachenko, V.. Non-self-adjoint periodic Dirac operators with finite-band spectra. Int. Equ. Oper. Theory 36 (2000), 325–348.Google Scholar
Yang, C.-Fu and Yurko, V.. Recovering Dirac operator with nonlocal boundary conditions. J. Math. Anal. Appl. 440 (2016), 156–166.CrossRefGoogle Scholar
Yurko, V. A., Method of spectral mappings in the inverse problem theory, Inverse and Ill-posed Problems Series (VSP, Utrecht, 2002).Google Scholar
Yurko, V. A.. Introduction to the theory of inverse spectral problems (Fizmatlit, Moscow, 2007) (in Russian).Google Scholar
Yurko, V. A.. Inverse spectral problems for differential operators and their applications (Gordon and Breach Science Publishers, Amsterdam, 2000) (in Russian).Google Scholar
Yurko, V. A.. An inverse spectral problem for singular non-selfadjoint differential systems. Sbornik: Math. 195 (2004), 1823–1854.Google Scholar
Yurko, V. A.. Inverse spectral problems for differential systems on a finite interval. Results Math. 48 (2005), 371–386.CrossRefGoogle Scholar
Yurko, V. A.. An inverse problem for differential systems on a finite interval in the case of multiple roots of the characteristic polynomial. Differ. Eqns. 41 (2005), 818–823.Google Scholar
Yurko, V. A.. An inverse problem for differential systems with multiplied roots of the characteristic polynomial. J. Inv. Ill-Posed Probl. 13 (2005), 503–512.CrossRefGoogle Scholar