curves of quantum spin systems can be obtained with very high accuracy [@Hotta2; @Hotta]. Besides, very recently, a further relation between the SSD and CFT, and its relevance to string theory is also pointed out in Refs. [@Tada:2014kza; @Ishibashi]

A key to understanding the ground-state correspondence between the SSD and uniform systems is that each chiral part of the SSD Hamiltonian annihilates the many-body ground state of the uniform system with PBC [@MaruyamaKH2011]. Here, we remark that the plane-wave representation of the single-particle basis, which is a suitable description for the uniform system with PBC, plays a very essential role in verifying the above statement. However, as shown numerically in Ref. [@ShibataH2011; @Hotta2; @Hotta], low-lying single-particle states in the SSD system seem to be localized around the center or edges of the chain, which would be rather puzzling. A natural question is: “What is the unified view of the physics behind the plane-wave description and the single-particle eigenstates of the SSD?" In this paper, we explore this question by examining the SSD of the free-fermion system defined in 1D continuous space. To this end, we formulate the SSD problem as the inverse problem of the quantum mechanical problem of a 