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.

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We consider Schrödinger operators on noncompact star-shaped graphs. The following topics will be treated under suitable decay conditions on the potential: characterization of the set of eigenvalues and expression of each eigen-projection; low-energy behavior of the resolvent, spectral representations of the operator restricted on in nite rays; determination of the Marchenko equations and the inverse scattering problems.

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