Abstract
We study the spectral properties of the Dirac operator L P,U generated in the space (L 2[0, π])2 by the differential expression By′ + P(x)y and by Birkhoff regular boundary conditions U, where y = (y 1, y 2)t, \(B = \left( {\begin{array}{*{20}{c}} { - i}&0 \\ 0&i \end{array}} \right)\), and the entries of the matrix P are complexvalued Lebesgue measurable functions on [0, π]. We also study the asymptotic properties of the eigenvalues {λ n } n∈Z of the operator L P,U as n → ∞ depending on the “smoothness” degree of the potential P; i.e., we consider the scale of Besov spaces B θ1,∞ , θ ∈ (0, 1). In the case of strongly regular boundary conditions, we study the asymptotic behavior of the system of normalized eigenfunctions of the operator L P,U , and in the case of regular but not strongly regular boundary conditions, we find the asymptotics of two-dimensional spectral projections.
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References
Levitan, B.M. and Sargsyan, I.S., Operatory Shturma–Liuvillya i Diraka (Sturm–Liouville and Dirac Operators), Moscow: Nauka, 1988.
Djakov, P. and Mityagin, B., Riesz Bases Consisting of Root Functions of 1D Dirac Operators, Proc. Amer. Math. Soc., 2013, vol. 141, pp. 1361–1375.
Burlutskaya, M.Sh., Kurdyumov, V.P., and Khromov, A.P., Refined Asymptotic Formulas for the Eigenvalues and Eigenfunctions of the Dirac System, Dokl. Ross. Akad. Nauk, 2012, vol. 443, no. 4, pp. 414–417.
Burlutskaya, M.Sh., Kornev, V.V., and Khromov, A.P., Dirac System with Nondifferentiable Potential and Periodic Boundary Conditions, Zh. Vychisl. Mat. Mat. Fiz., 2012, vol. 52, no. 9, pp. 1621–1632.
Kornev, V.V. and Khromov, A.P., Dirac System with Nondifferentiable Potential and Antiperiodic Boundary Conditions, Izv. Sarat. Univ. Nov. Ser. Mat. Mekh. Inform., 2013, vol. 13, no. 3, pp. 28–35.
Savchuk, A.M. and Shkalikov, A.A., The Dirac Operator with Complex–Valued Integrable Potential, Math. Notes, 2014, vol. 96, no. 5, pp. 3–36.
Savchuk, A.M. and Sadovnichaya, I.V., Riesz Basis Property with Parentheses for Dirac System with Integrable Potential, Sovrem. Mat. Fundam. Napravl., 2015, vol. 58, pp. 128–152.
Savchuk, A.M. and Sadovnichaya, I.V., Asymptotic Formulas for Fundamental Solutions of the Dirac System with Complex-Valued Integrable Potential, Differ. Uravn., 2013, vol. 49, no. 5, pp. 573–584.
Bennett, C. and Sharpley, R., Interpolation of Operators, Boston, 1988.
Triebel, H., Interpolation Theory, Function Spaces, Differential Operators, Berlin: Birkhäuser, 1977. Translated under the title Teoriya interpolyatsii, funktsional’nye prostranstva, differentsial’nye operatory, Moscow: Mir, 1980.
Bergh, J. and Löfsrtöm, J., Interpolation Spaces. An Introduction, Berlin–New York: Springer-Verlag, 1976. Translated under the title Interpolyatsionnye prostranstva. Vvedenie, Moscow: Mir, 1980.
Coddington, E. and Levinson, N., Theory of Ordinary Differential Equations, New York, 1955. Translated under the title Teoriya obyknovennykh differentsial’nykh uravnenii, Moscow: Inostrannaya Literatura, 1958.
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Original Russian Text © A.M. Savchuk, 2016, published in Differentsial’nye Uravneniya, 2016, Vol. 52, No. 4, pp. 454–469.
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Savchuk, A.M. Dirac system with potential lying in Besov spaces. Diff Equat 52, 431–446 (2016). https://doi.org/10.1134/S0012266116040042
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DOI: https://doi.org/10.1134/S0012266116040042