  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.
