On Schrödinger Operators Modified by \(\delta\) Interactions

We study the spectral properties of a Schr\"{o}dinger operator \(H_0\) 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 \(H_0\). 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.