In this paper, we analyze the non-selfadjoint Sturm-Liouville operator L defined in the Hilbert space L 2 (R, H) of vector-valued functions which are strongly-measurable and square-integrable in R. L is defined L(y) = −y ′′ + Q(x)y, x ∈ R, for every y ∈ L 2 (R, H) where the potential Q(x) is a non-selfadjoint, completely continuous operator in a separable Hilbert space H for each x ∈ R. We obtain the Jost solutions of this operator and examine the analytic and asymptotic properties. Moreover, we find the point spectrum and the spectral singularities of L and also obtain the sufficient condition which assures the finiteness of the eigenvalues and spectral singularities of L.

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In this paper, we consider the boundary value problem [Formula: see text][Formula: see text] where λ is the spectral parameter and [Formula: see text] is a Hermitian matrix such that [Formula: see text] and αi ∈ ℂ, i = 0, 1, 2, with α2 ≠ 0. In this paper, we investigate the eigenvalues and spectral singularities of L. In particular, we prove that L has a finite number of eigenvalues and spectral singularities with finite multiplicities, under the Naimark and Pavlov conditions.

In this paper, we consider the discrete Sturm-Liouville operator generated by second order difference equation with non-selfadjoint operator coefficient. This operator is the discrete analogue of the Sturm-Liouville differential operator generated by Sturm-Liouville operator equation which has been studied in detail. We find the Jost solution of this operator and examine its asymptotic and analytical properties. Then, we find the continuous spectrum, the point spectrum and the set of spectral singularities of this discrete operator. We finally prove that this operator has a finite number of eigenvalues and spectral singularities under a specific condition.

In this paper, we consider the second order di¤erence equation de…ned on the whole axis with selfadjoint operator coe¢ cients. The main objective of this study is to obtain the continuous and discrete spectrum of the discrete operator which is generated by this di¤erence equation. To achieve this, we …rst obtain the Jost solutions of this equation explicitly and then examine the analytical and asymptotic properties of these solutions. With the help of these properties, we …nd the continuous and discrete spectrum of this operator. Finally we obtain a su¢ cient condition which ensures that this operator has a …nite number of eigenvalues.