Unconditional convergence of spectral decompositions of 1D Dirac operators with regular boundary conditions

By Plamen Djakov, Boris Mityagin. 2010

84 citations recorded by the source corpus.

One-dimensional Dirac operators L-bc(v)y = i(1 0 0 -1) dy/dx + v(x)y, y = (y(1)y(2)), x is an element of [0, pi], considered with L-2-potentials v (x) = (0 Q(x) P(x)0) and subject to regular boundary conditions (bc), have discrete spectrum. For strictly regular bc, it is shown that every eigenvalue of the free operator L-bc(0) is simple and has the form lambda(0)(k,alpha) = k + tau(alpha), where alpha is an element of {1, 2}, k is an element of 2Z and tau(alpha) = tau(alpha)(bc); if vertical bar k vertical bar > N(v, bc), each of the discs D-k(alpha) = {z : vertical bar z - lambda(0)(k,alpha) vertical bar < rho = rho(bc)}, alpha is an element of {1, 2}, contains exactly one simple eigenvalue lambda(k,alpha)< of L-bc (v) and (lambda(k,alpha) - lambda(0)(k,alpha))(k is an element of 2Z) is an l(2)-sequence. Moreover, it is proven that the root projections P-n,P-alpha = 1/(2 pi i) integral(partial derivative Dn)alpha (z - L-bc (v))(-1) dz satisfy the Bari-Markus condition Sigma(vertical bar n vertical bar>N) parallel to P-n,P-alpha - P-n,alpha(0)parallel to(2) <infinity, n is an element of 2Z, where P-n,alpha(0) are the root projections of the free operator L-bc(0). Hence, for strictly regular bc, there is a Riesz basis consisting of root functions (all but finitely many being eigenfunctions). Similar results are obtained for regular but not strictly regular bc-then in general there is no Riesz basis consisting of root functions, but we prove that the corresponding system of two-dimensional root projections is a Riesz basis of projections.

Subfields: Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6d78c73387411773b87a8a7f93d30825e9318b46

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Citations in the corpus

• Dirac operator with complex-valued summable potential (2014, unknown): “2, большей частью содержатся в статье [5] Джакова и Митягина.”
• On the Riesz basis property of root vectors system for $2 \times 2$ Dirac type operators (2015, methodology): “Emphasize that methods used in [9, 3] essentially use condition Q ∈ L2[0, 1] ⊗ C2×2 (i.”
• On the completeness of root subspaces of boundary value problems for first order systems of ordinary differential equations (2012, background): “Moreover, in [10] they proved the Riesz basis property of the system of EAF for BVP with general strictly regular BC under the assumption Q12,Q21 ∈ L2[0,1].”
• Criteria for existence of Riesz bases consisting of root functions of Hill and 1D Dirac operators (2011, unknown): “Let us notice that we proved (79)–(80) for arbitrary regular boundary condition – see Theorems 15 and 20 in [16]; however, we do not use these results from [16] in the present paper.”