Dirac operator with complex-valued summable potential

Completeness Property of One-Dimensional Perturbations of Normal and Spectral Operators Generated by First Order Systems

“…Remark 3.14. (i) The Riesz basis property for 2 × 2 Dirac operators L U 1 ,U 2 has been investigated in numerous papers (see [41,8,6,9,11,12,18,24,27,39] and references therein). The most complete result was recently obtained independently and by different methods in [24,27] and [39].…”

“…(i) The Riesz basis property for 2 × 2 Dirac operators L U 1 ,U 2 has been investigated in numerous papers (see [41,8,6,9,11,12,18,24,27,39] and references therein). The most complete result was recently obtained independently and by different methods in [24,27] and [39]. Namely, assuming that B = B * and Q(·) ∈ L 1 ([0, 1]; C 2×2 ) it is proved in [24,27] (the general case of b 1 b 2 < 0) and [39] (the Dirac case, b 1 = −b 2 ) that the system of root vectors of equation (1.3) subject to regular boundary conditions constitutes a Riesz basis with parentheses in L 2 ([0, 1]; C 2 ) and ordinary Riesz basis provided that BC are strictly regular.…”

“…(ii) Note also that an important role in proving Riesz basis property in [24,27] and [39] is playing the following asymptotic formula λ n = λ 0 n + o(1), as n → ∞, n ∈ Z, (3.36) for the eigenvalues {λ n } n∈Z of the operator L C,D (B, Q) with regular BC (and summable potential matrix Q), where {λ 0 n } n∈Z is the sequence of eigenvalues of the unperturbed operator L C,D (B, 0). Note also that formula (3.36) has recently been applied to investigation of spectral properties of Dirac systems on star graphs [1].…”

“…Besides, P. Djakov and B. Mityagin [10] imposing certain smoothness condition on Q(·) proved equiconvergence of the spectral decompositions for 2 × 2 Dirac equations subject to general regular boundary conditions. Moreover, the Riesz basis property for 2 × 2 Dirac operators L U 1 ,U 2 has been investigated in numerous papers (see [41,8,6,9,11,12,18,24,27,39] and references therein, and discussion in Remark 3.14).…”

Completeness Theorem for Eigenparameter Dependent Dissipative Dirac Operator with General Transfer Conditions

“…For self-adjoint Sturm-Liouville operators, this kind of problems have been investigated by Mukhtarov [16,17,19]. It should be noted that completeness properties and the Riesz basis property for strongly regular boundary value problems for Dirac operators on a finite interval have been established in [33][34][35].…”