## Abstract

We study the Hamiltonians H_{X,α,q} with δ-type point interactions at the centers x_{k} on the positive half line in terms of energy forms. We establish analogs of some classical results on operators H_{q}=-d^{2}/dx^{2}+q with locally integrable potentials q∈L_{loc}^{1}(R_{+}). In particular, we prove that the Hamiltonian H_{X,α,q} is self-adjoint if it is lower semibounded. This result completes the previous results of Brasche ["Perturbation of Schrödinger Hamiltonians by measures-selfadjointness and semiboundedness," J. Math. Phys.26, 621 (1985)] on lower semiboundedness. Also we prove the analog of Molchanov's discreteness criteria, Birman's result on stability of a continuous spectrum, and investigate discreteness of a negative spectrum. In the recent paper [Kostenko, A. and Malamud, M., "1-D Schrödinger operators with local point interactions on a discrete set," J. Differ. Equations249, 253 (2010)], it was shown that the spectral properties of H_{X,α}{colon equals}H_{X,α,0} correlate with the corresponding spectral properties of a certain class of Jacobi matrices. We apply the above mentioned results to the study of spectral properties of these Jacobi matrices.

Original language | English |
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Article number | 102102 |

Journal | Journal of Mathematical Physics |

Volume | 51 |

Issue number | 10 |

DOIs | |

Publication status | Published - Oct 2010 |

Externally published | Yes |