Spectral properties and an inverse eigenvalue problem for non-symmetric systems of ordinary differential operators

Abstract

We consider a nonsymmetric first-order differential operator (Au)(x) = 0 1 1 0 (du/dx)(x) + P (x)u(x), 0 < x < 1, where P is a 2 × 2 matrix whose components are in L 2 (0, 1). We study an eigenvalue problem for A with boundary conditions at x = 0, 1. We establish an asymptotic form of the eigenvalues and prove that the set of the root vectors forms a Riesz basis in {L 2 (0, 1)} 2 . Moreover we show some characterization of L 2 -coefficients in an inverse eigenvalue problem. The key is a transformation formula.

On the completeness of root subspaces of boundary value problems for first order systems of ordinary differential equations

“…In connection with Corollary 1.3 and Proposition 1.4 we mention the papers [49,50,21,39] and [6][7][8][9][10][11][12][13], that appeared during the last decade. Basically they are devoted to Riesz basis property of EAF for BVP with strictly regular (and just regular) BC for 2 × 2 Dirac systems.…”

“…(ii) In connection with Theorem 5.1 and other results of this section we mention the papers [49,50,21] devoted to the Riesz basis property of EAF for BVP with separated (and hence strictly regular) BC for 2 × 2 Dirac systems [49,50,39] and for 2 × 2 Dirac type systems [21].…”

Bari–Markus property for Riesz projections of 1D periodic Dirac operators

“…For Dirac operators (2.1) the results on unconditional convergence are sparse and not complete so far [13,14,18,19,[30][31][32].…”

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].…”