Point scatterers and resonances in low number of dimensions

It is well known that in 1D the cross section of a point scatterer increases along with the scatterer's strength (potential). In this paper we show that this is an exceptional case, and for lower or higher number of dimensions, i.e., 0<d<1 and 1<d≤2 (d is the dimensions number), the cross section does not increase monotonically with the scatterer's strength. In fact, the cross section exhibits a resonance dependence on the scatterer's strength, and in the singular 2D case it gets its maximum value for an infinitely weak strength. We use this fact to show that two totally different generalized function can describe exactly the same physical entity (the same scatterer).