1-D Schrödinger operators with local point interactions on a discrete set
By A. Kostenko, M. Malamud. 2010
150 citations recorded by the source corpus.
Abstract Spectral properties of 1-D Schrodinger operators H X , α : = − d 2 d x 2 + ∑ x n ∈ X α n δ ( x − x n ) with local point interactions on a discrete set X = { x n } n = 1 ∞ are well studied when d ∗ : = inf n , k ∈ N | x n − x k | > 0 . Our 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.
Subfields: Computational Theory and Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/e7aa339c9a7da02debc454d10cd357382745a33c
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In Computational Theory and Mathematics
In Mathematical Physics
Citations in the corpus
- Boundary pairs associated with quadratic forms (2012, background): “1 Recently, boundary triple methods have also been used in [52] for the spectral analysis of Laplacians with infinitely many delta-interactions in dimension 1 which become arbitrarily close.”
- Boundary Value Problems, Weyl Functions, and Differential Operators (2020, background): “However, for a selection of papers in which boundary triplet techniques were applied to differential operators and other related problems we refer to [28, 50, 86, 87, 97, 112, 113, 123, 143, 144, 155, 169, 170, 173, 174, 184, 185, 241, 288, 307, 342, 343, 344, 377, 461, 475, 477, 478, 483, 511, 529,”