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Abstract. The concepts of boundary relations and the corresponding Weyl families are introduced. Let S be a closed symmetric linear operator or, more generally, a closed symmetric relation in a Hilbert space H, let H be an auxiliary Hilbert space, let

MSC: 34L05 34L40 47E05 47B25 47B36 81Q10 Keywords: Schrödinger operator Local point interaction Self-adjointness Lower semiboundedness Discreteness Spectral properties of 1-D Schrödinger operators H X,αOur paper is devoted to the case d * = 0. We consider H X,α in the framework of extension theory of symmetric operators by applying the technique of boundary triplets and the corresponding Weyl functions. We show that the spectral properties of H X,α like self-adjointness, discreteness, and lower semiboundedness correlate with the corresponding spectral properties of certain classes of Jacobi matrices. Based on this connection, we obtain necessary and sufficient conditions for the operators H X,α to be self-adjoint, lower semibounded, and discrete in the case d * = 0. The operators with δ -type interactions are investigated too. The obtained results demonstrate that in the case d * = 0, as distinguished from the case d * > 0, the spectral properties of the operators with δ-and δ -type interactions are substantially different.

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