Point scatterers in low number of dimensions

TL;DR

This paper demonstrates that in dimensions other than 1D, the cross section of a point scatterer does not monotonically increase with strength but exhibits resonance behavior, with maximum at weak strength in 2D.

Contribution

It reveals the non-monotonic and resonant dependence of scatterer cross section on strength in dimensions 0<d<=2, contrasting with the 1D case.

Findings

Cross section does not increase monotonically with scatterer strength in 0<d<=2.

Resonance dependence of cross section on scatterer strength.

Maximum cross section occurs at infinitesimally weak strength in 2D.

Abstract

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 in all the other cases, where a point defect has a physical meaning, 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 functions can describe exactly the same physical entity (the same scatterer).