On Schr\"{o}dinger 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.