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The iterative solution of the Schrodinger integral equation (for zero energy) associated with certain singular potentials containing logarithmic singularities of the form lognr/r2 fails to converge unless a cut-off is introduced. The solution for these potentials is shown to be renormalizable by a multiplicative factor dependent upon the cut-off. The existence of a perturbation-series solution with divergent terms as an indication that the closed solution possesses an essential singularity in the expansion parameter is examined for a soluble case, and a functional relationship between this parameter and the cut-off is found which explicitly exhibits the essential singularity in the perturbation series. This phenomenon is conjectured to occur in a more general situation.

The infinite sequence of conserved quantities which were recently discovered in free-field theories by Kibble, are shown to be a consequence of translation invariance and the quadratic nature of the Lagrangian. This observation leads to a natural extension of Kibble’s work by exploiting the full Lorentz invariance of the free-field equations, leading to the construction of further new conservation laws. The question whether the existence of a conservation law implies an invariance of the Lagrangian is raised and answered in the negative.