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

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Official URL: http://dx.doi.org/10.1512/iumj.2012.61.4531

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.