2010/08/24 by Plamen Djakov, Djakov, Plamen, Boris Mityagin +1
Mathematics · #34L10 #34L40 #47E05 #FOS: Mathematics #Spectral Theory (math.SP) #math.SP #msc:34L10 #msc:34L40 #msc:47E05
paper · pdf · doi:10.48550/arxiv.1008.4095
arxiv created 2010/08/24 · arxiv updated 2010/08/25
One dimensional Dirac operators Lbc(v) y = i \beginpmatrix 1 0 0 -1 \endpmatrix (dy)/(dx) + v(x) y, y = \beginpmatrix y1 y2 \endpmatrix, x∈[0,π], considered with L2-potentials v(x) = \beginpmatrix 0 & P(x) Q(x) & 0 \endpmatrix and subject to regular boundary conditions (bc), have discrete spectrum. For strictly regular bc, it is shown that every eigenvalue of the free operator L0bc is simple and has the form λk,α0 = k + τα where α∈ \1,2\, k ∈ 2 ℤ and τα=τα(bc); if |k|>N(v, bc) each of the discs Dkα= \z: |z-λk,α0| <ρ=ρ(bc) \ , α∈ \1,2\, contains exactly one simple eigenvalue λk,α of Lbc (v) and (λk,α -λk,α0)k∈ 2ℤ is an ℓ2 -sequence. Moreover, it is proven that the root projections Pn,α = (1)/(2πi) ∫∂ Dαn (z-Lbc (v))-1 dz satisfy the Bari--Markus condition ∑|n| > N ‖Pn,α - Pn,α0‖2 < ∞, n ∈ 2ℤ, where Pn0 are the root projections of the free operator L0bc. 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.