a determinant of single-particle wavefunctions $\varphi (x)$ which are determined by the Schrödinger-like equation: $$-f(x) \left( \frac{d^2}{dx^2} +\mu \right)\varphi(x) = \xi \varphi(x),
\label{ssdh}$$ where $\xi$ denotes the eigenvalue. Note that the Hamiltonian here is not self-adjoint in the standard sense and the appropriate inner product for the SSD problem will be introduced in section 5. For later convenience, we assume either periodic or antiperiodic boundary condition for $\varphi (x)$, depending on the total number of fermions: $\varphi (x+L) = (-1)^N \varphi (x)$. We denote by $\{\puni_n(x)\}$ and $\{\pssd_n(x)\}$ the sets of single-particle eigenfunctions of the uniform ($f(x)=1$) and SSD ($f(x)=\sin^2 \frac{\pi}{L}x$) systems, respectively. For the uniform system, $\{\puni_n(x)\}$ is of course described by the plane waves and is easy to obtain. For periodic boundary conditions, one finds that the linear space of $\{ \puni_n (x) \}$ is spanned by $$\exp \left( \pm i \frac{2 n \pi}{L}x \right), 
\quad (n=0,1,2, ...),$$ and for antiperiodic boundary conditions, $$\exp \left( \pm i \frac{(2 n+1) \pi}{L}x \right), 
\quad (n=0,1,2, ...).$$ In contrast, to obtain $\{ \pssd_n (x) \}$ for the SSD problem is an intriguing problem, because the translational symmetry is explicitly broken by the position-dependent factor $f(x)$. However, for specific values of the 