problem with SSD as the inverse problem of the $1/\sin^2$ potential problem. Then we obtain the single-particle eigenstates for specific values of the chemical potentials. In section 4, we prove the ground-state correspondence between the uniform and SSD systems by showing that the single-particle eigenstates for the SSD can be expressed as linear combinations of plane waves. In section 5, we briefly comment on the single-particle states with positive energy in the SSD system. In section 6, we compare the SSD in the continuous space with that in a lattice, and discuss the role of the geometric correction. In the last section, we summarize the results obtained and discuss the prospects for future researches.

Definition of the SSD problem in a one-dimensional ring {#sec:problem_setup}
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We start with the definition of the SSD problem in 1D continuous space. Consider $N$ fermions on a ring of length $L$. We denote by ${\hat c}^\dagger (x)$ and ${\hat c}(x)$ the fermionic field operators at position $x$ ($0 \le x \le L$). They satisfy the canonical anticommutation relations: $\{ {\hat c}(x), {\hat c}^\dagger (y) \} = \delta (x-y)$, $\{ {\hat c}(x), {\hat c} (y) \} =\{ {\hat c}^\dagger (x), {\hat c}^\dagger (y) \}=0$. In 