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In this paper we study the scattering length for positive additivefunctionals of symmetric stable processes on ${\bf R}^d$. The additivefunctionals considered here are not necessarily continuous. We prove that thesemi-classical limit of the scattering length equals the capacity of thesupport of a certain measure potential, thus extend previous results for thecase of positive continuous additive functionals. We also give an equivalentcriterion for the fractional Laplacian with a measure valued non-local operatoras a perturbation to have purely discrete spectrum in terms of the scatteringlength, by considering the connection between scattering length and the bottomof the spectrum of Schr\"odinger operator in our settings.

On a scattering length for additive functionals and spectrum of fractional Laplacians with non-local perturbations