On Schrodinger operators modified by δ interactions

We study the spectral properties of a Schrodinger operator H0 modified by delta interactions and show explicitly how the poles of the new Green's function are rearranged relative to the poles of original Green's function of H0. We prove that the new bound state energies are interlaced between the old ones, and the ground state energy is always lowered if the delta interaction is attractive. We also derive an alternative perturbative method of finding the bound state energies and wave functions under the assumption of a small coupling constant in a somewhat heuristic manner. We further show that these results can be extended to cases in which a renormalization process is required. We consider the possible extensions of our results to the multi center case, to delta interaction supported on curves, and to the case, where the particle is moving in a compact two-dimensional manifold under the influence of delta interaction. Finally, the semi-relativistic extension of the last problem has been studied explicitly.(c) 2023 Elsevier Inc. All rights reserved.