On the spectrum of non-selfadjoint Dirac operators with two-point boundary conditions

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Abstract

We consider spectral problem for the Dirac operator with arbitrary two-point boundary conditions and any square integrable potential . The necessary and sufficient conditions are established that an entire function must satisfy in order to be a characteristic determinant of the specified operator. In the case of irregular boundary conditions, conditions are found under which a set of complex numbers is the spectrum of the problem under consideration.

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About the authors

A. S. Makin

Peoples’ Friendship University of Russia

Author for correspondence.
Email: alexmakin@yandex.ru
Russian Federation, Moscow

References

  1. Albeverio, A. Inverse spectral problems for Dirac operators with summable potentials / A. Albeverio, R. Hryniv, Ya. Mykytyuk // Russ. J. Math. Phys. — 2005. — V. 12, № 4. — P. 406–423.
  2. Lunyov A. On the Riesz basis property of root vectors system for 2×2 Dirac type operators / A. Lunyov, M. Malamud // J. Math. Anal. Appl. — 2016. — V. 441, № 1. — P. 57–103.
  3. Savchuk, A.M. The Dirac operator with complex-valued summable potential / A.M. Savchuk, A.A. Shkalikov // Math. Notes. — 2014. — V. 96, № 5. — P. 777–810.
  4. Savchuk, A.M. Spectral analysis of one-dimensional Dirac system with summable potential and Sturm–Liouville operator with distribution coefficients / A.M. Savchuk, I.V. Sadovnichaya // Contemporary Mathematics. Fundamental Directions. — 2020. — V. 66, № 3. — P. 373–530.
  5. Misyura, T.V. Characterization of spectra of periodic and anti-periodic problems generated by Dirac’s operators. II / T.V. Misyura // Theoriya functfii, funct. analiz i ikh prilozhen. — 1979. — V. 31. — P. 102–109. [in Russian]
  6. Nabiev, I.M. Solution of the quasiperiodic problem for the Dirac system / I.M. Nabiev // Math. Notes. —2011. — V. 89, № 6. — P. 845–852.
  7. Djakov, P. Unconditional convergence of spectral decompositions of 1D Dirac operators with regular boundary conditions / P. Djakov, B. Mityagin // Indiana Univ. Math. J. — 2012. — V. 61. — P. 359–398.
  8. Yurko, V.A. Inverse spectral problems for differential systems on a finite interval / V.A. Yurko // Results in Mathematics. — 2005. — V. 48, № 3–4. — P. 371–386.
  9. Makin, A.S. On the spectrum of two-point boundary value problems for the Dirac operator / A.S. Makin // Differ. Equat. — 2021. — V. 57, № 8. — P. 993–1002.
  10. Tkachenko, V. Non-self-adjoint periodic Dirac operators / V. Tkachenko // Oper. Theory: Adv. and Appl. —2001. — V. 123. — P. 485–512.
  11. Marchenko, V.A. Sturm–Liouville operators and their applications / V.A. Marchenko. — Basel : Birkhauser Verlag, 1986.
  12. Tkachenko, V. Non-self-adjoint periodic Dirac operators with finite-band spectra / V. Tkachenko // Int. Equ. Oper. Theory. — 2000. — V. 36. — P. 325–348.
  13. Levin, B.Ya. Lectures on Entire Functions / B.Ya. Levin . — Providence : American Mathematical Society, 1996.
  14. Levin, B.Ya. On small perturbations of the set of zeros of functions of sine type / B.Ya. Levin, I.V. Ostrovskii // Math. USSR-Izv. — 1980. — V. 14, № 1. — P. 79–101.
  15. Lavrentiev, M.A. Methods of Theory of Complex Variable / M.A. Lavrentiev, B.V. Shabat. — Moscow : Nauka, 1973. — 736 p. [in Russian]
  16. Sansug, J.-J. Characterization of the periodic and antiperiodic spectra of nonselfadjoint Hill’s operators / J.-J. Sansug, V. Tkachenko // Oper. Theory Adv. and Appl. — 1997. — V. 98. — P. 216–224.
  17. Nikolskii, S.M. Approximation of Functions of Several Variables and Embedding Theorems / S.M. Nikolskii. —Moscow : Nauka, 1977. — 456 p. [in Russian]

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