Spectral Theory of Semibounded Sturm-Liouville Operators with Local Interactions on a Discrete Set
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We study the Hamiltonians HX,a,q with -type point interactions at the centers xk on the positive half line in terms of energy forms. We establish analogs of some classical results on operators Hq=−d2 /dx2+q with locally integrable potentials q Lloc1 R+. In particular, we prove that the Hamiltonian HX,,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. Equations 249, 253 2010, it was shown that the spectral properties of HX,ªHX,,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.
Alveverio, S., Kostenko, A., Malamud, M.:Spectral Theory of Semibounded Sturm-Liouville Operators with Local Interactions on a Discrete Set.Journal of Mathematical Physics J. Math. Phys. 51, 102102 (2010).