Riesz Bases of Root Vectors of Indefinite Sturm-liouville Problems with Eigenparameter Dependent Boundary Conditions. Ii

We employ an operator theoretic setting established in [2]. Under Condition 2.1 below, a self-adjoint (actually quasi-uniformly positive [7]) operator A in the Krein space L2,r(−1, 1)⊕C 2 ∆ is associated with the eigenvalue problem (1.1), (1.2). Here ∆ is a 2 × 2 nonsingular Hermitean matrix which is determined by M and N; see Section 2 for details. We remark that the topology of this Krein space is that of the corresponding Hilbert space L2,|r| ⊕ C 2 |∆|. Here and in the rest of the paper we abbreviate L2,r(−1, 1) to L2,r and L2,|r|(−1, 1) to L2,|r|. Our main goal in this paper is to provide sufficient conditions on the coefficients in (1.1), (1.2) under which there is a Riesz basis of the above Hilbert space consisting of the union of bases for all the root subspaces of the above operator A. This will be referred to for the remainder of this section as the Riesz basis property of A.