Spectral theory of semibounded Sturm-Liouville operators with local interactions on a discrete set

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=-d²/dx²+q with locally integrable potentials q∈L_loc¹(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.