Abstract
This paper considers potentials that are reflectionless for a range of energies and not reflectionless elsewhere. The focus of this paper is to see if reflection coefficients of this type have characteristics that would make it possible to solve for the associated potentials using known methods. Using the Gelfand-Levitan equation it is shown that the spectral measure function has several limits that match well to known solved cases. Thus, it seems that pursuing cases of potentials of this type is reasonable.
Highlights
Inverse scattering on the line is a well-known method to determine the potential V(r) given the reflection coefficient R(k)
Using the Gelfand-Levitan equation it is shown that the spectral measure function has several limits that match well to known solved cases
This is done by using the Gelfand-Levitan Equation [1,2,3]: (1)
Summary
Inverse scattering on the line is a well-known method to determine the potential V(r) given the reflection coefficient R(k). This is typically done by successive iteration [4]. A case of particular interest is that of reflectionless potentials [5,6,7,8] These are cases where the reflection coefficient is zero but the potential is non-zero. This paper considers the case of a reflection coefficient that is zero in a region and non-zero elsewhere. This would correspond to a potential that did not reflect a wave for a range of energies but did in general reflect outside of that range. The goal of this paper is to see what particular characteristics these potentials have that can aid in calculating corresponding potentials
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