In this thesis the one-dimensional complex scaling method for generalized scaling profiles is introduced and applied to equations of Helmholtz-type with well known solutions in the exterior domain. We discuss the meaning of radiating solutions, prove conditions for the existence of unique radiating solutions to scattering problems, and show that solutions to scaled and unscaled problems are equivalent in some sense. We transfer most of these results to the time-independent Schrödinger equation with quadratically decaying potential functions. Potential functions of this type are non-trivial in the exterior domain and lead to solutions with inexplicit asymptotic behavior. We analyze the discrete spectrum of the associated Hamilton operator and test its dependence on the parameter a as well as the influence of discretization parameters. The method is implemented with the finite element software NGSolve.