Generalized Supersymmetric Quantum Mechanics and Reflectionless Fermion Bags in 1+1 Dimensions

TL;DR

This paper revisits the reflectionless nature of static fermion bags in 1+1 dimensional models, critiquing previous asymptotic analyses and connecting the spectral properties to generalized supersymmetric quantum mechanics.

Contribution

It provides a critique of earlier asymptotic methods and links the spectral theory of Dirac operators in fermion bags to generalized supersymmetric quantum mechanics.

Findings

Previous asymptotic analysis was incomplete.

Reflectionless property of fermion bags is supported by supersymmetric quantum mechanics.

The spectral theory of Dirac operators is connected to generalized supersymmetry.

Abstract

We study static fermion bags in the 1+1 dimensional Gross-Neveu and Nambu-Jona-Lasinio models. It has been known, from the work of Dashen, Hasslacher and Neveu (DHN), followed by Shei's work, in the 1970's, that the self-consistent static fermion bags in these models are reflectionless. The works of DHN and of Shei were based on inverse scattering theory. Several years ago, we offered an alternative argument to establish the reflectionless nature of these fermion bags, which was based on analysis of the spatial asymptotic behavior of the resolvent of the Dirac operator in the background of a static bag, subjected to the appropriate boundary conditions. We also calculated the masses of fermion bags based on the resolvent and the Gelfand-Dikii identity. Based on arguments taken from a certain generalized one dimensional supersymmetric quantum mechanics, which underlies the spectral theory of these Dirac operators, we now realize that our analysis of the asymptotic behavior of the resolvent was incomplete. We offer here a critique of our asymptotic argument.