Piecewise-constant potentials with practically the same fixed-energy phase shifts

TL;DR

This paper demonstrates that different positive, compactly supported spherically symmetric potentials can produce nearly identical fixed-energy phase shifts, challenging previous assumptions about their uniqueness.

Contribution

It provides explicit examples of such potentials and introduces a hybrid stochastic-deterministic method for their construction.

Findings

Different potentials can produce almost identical phase shifts.

Explicit examples of positive potentials with similar phase shifts are constructed.

A hybrid method for potential reconstruction is developed.

Abstract

It has recently been shown that spherically symmetric potentials of finite range are uniquely determined by the part of their phase shifts at a fixed energy level $k^2>0$. However, numerical experiments show that two quite different potentials can produce almost identical phase shifts. It has been guessed by physicists that such examples are possible only for "less physical" oscillating and changing sign potentials. In this note it is shown that the above guess is incorrect: we give examples of four positive spherically symmetric compactly supported quite different potentials having practically identical phase shifts. The note also describes a hybrid stochastic-deterministic method for global minimization used for the construction of these potentials.