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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-adjointHamiltonian operator representing the negative Laplacian on the positivehalf-line with a Dirichlet (resp. Neuman) boundary condition at the originperturbed by an attractive Dirac distribution $-\lambda \delta(x-x_0)$ for anyfixed 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 theanalytic continuation of the resolvent. These results will be connected withthe study of the ground state energy of two remarkable three-dimensionalself-adjoint operators, studied in depth in Albeverio's monograph, perturbed byan attractive $\delta$-distribution supported on the spherical shell of radius$r_0$.

The Schr\"odinger particle on the half-line with an attractive $\delta$-interaction: bound states and resonances