  In this paper we provide a detailed description of the eigenvalue $
E_{D}(x_0)\leq 0$ (respectively $ E_{N}(x_0)\leq 0$) of the self-adjoint
Hamiltonian operator representing the negative Laplacian on the positive
half-line with a Dirichlet (resp. Neuman) boundary condition at the origin
perturbed by an attractive Dirac distribution $-\lambda \delta(x-x_0)$ for any
fixed value of the magnitude of the coupling constant. We also investigate the
$\lambda$-dependence of both eigenvalues for any fixed value of $x_0$.
Furthermore, we show that both systems exhibit resonances as poles of the
analytic continuation of the resolvent. These results will be connected with
the study of the ground state energy of two remarkable three-dimensional
self-adjoint operators, studied in depth in Albeverio's monograph, perturbed by
an attractive $\delta$-distribution supported on the spherical shell of radius
$r_0$.
