In this work, an eigenvalue problem for Dirac operators with discontinuities is investigated in the case when an eigenparameter appears not only in the differential equation but also in the boundary and jump conditions, polynomially. Mathematics Subject Classification: 34A55, 34B24, 34L05

Form domains are characterized for regular 2n-th order differential equations subject to general self-adjoint boundary conditions depending affinely on the eigenparameter. Corresponding modes of convergence for eigenfunction expansions are studied, including uniform convergence of the first n− 1 derivatives.

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 spa...

and Applied Analysis 3 2. Jost Solution of 1.4 We will denote the solution of 1.4 satisfying the condition lim x→∞ y x, λ e−iλx 1, λ ∈ C : {λ : λ ∈ C, Imλ ≥ 0}, 2.1 by e x, λ . The solution e x, λ is called the Jost solution of 1.4 . Under the condition ∫∞ 0 x ∣ ∣q x ∣ ∣dx < ∞, 2.2 the Jost solution has a representation

We consider a regular indefinite Sturm-Liouville problem with two self-adjoint boundary conditions, one being affinely dependent on the eigen-parameter. We give sufficient conditions under which a basis of each root subspace for this Sturm-Liouville problem can be selected so that the union of all these bases constitutes a Riesz basis of a corresponding weighted Hilbert space.