The authors consider a

$\lambda $-linear boundary value problem for a singular second order differential expression

$\ell $ on the half-line

$(0,\infty )$, where it is assumed that both endpoints 0 and

$\infty $ are in the limit circle case. A dissipative boundary condition is imposed at 0 and the spectral parameter appears linearly in the boundary condition at

$\infty $. The

$\lambda $-dependent boundary condition at

$\infty $ is transformed into a constant boundary condition for a suitable operator

$A$ in

${L}^{2}\left((0,\infty )\right)\oplus \u2102$ which is a maximal dissipative extension of the minimal operator associated to

$\ell $ in

${L}^{2}\left((0,\infty )\right)$. By means of the minimal selfadjoint dilation of the maximal dissipative operator

$A$ it is shown that the eigenfunctions of the original boundary value problem are complete in

${L}^{2}\left((0,\infty )\right)$. Furthermore, the characteristic function of the maximal dissipative operator

$A$ is identified with the Lax-Phillips scattering matrix of the selfadjoint dilation.