N$), though the linear space of them is identical to that of the plane waves.

For the continuous SSD, the self-adjoint form of the differential equation tells us that an appropriate inner product for $\psi^{N-\beta}_\beta(x)$ is $$\langle\langle \psi^{N-\beta}_\beta,\, \psi^{N-\beta'}_{\beta'} \rangle\rangle := \frac{1}{L}  \int_0^L \frac{dx}{\sin^2\frac{\pi}{L}x}\,
\psi^{N-\beta}_\beta(x)\, \psi^{N-\beta'}_{\beta'}(x)  = I_{N,\beta}\, \delta_{\beta,\beta'},
\label{innerprod}$$ where the coefficient $I_{N,\beta}$ is given by $$I_{N,\beta} = 2^{2-2\beta}\, ({\cal N}^{N-\beta}_\beta)^2
\frac{(N+\beta-1)!}{(2\beta-1)\, (N-\beta)!\, [(\beta-1)!]^2},$$ the derivation of which is given in \[app:coeffI\]. The factor $\sin^2\frac{\pi}{L}x$ in the integrand in Eq. (\[innerprod\]) can be regarded as a geometric factor attributed to the spatial deformation. With this definition, the orthogonality of the single-particle wavefunctions for $\beta\ge 1$ is correctly recovered. Here, we should mention that $\psi_0^N(x) \propto \cos(\frac{N\pi}{L}x )$ for $\beta=0$ is not normalizable with Eq. (\[innerprod\]). We will discuss this problem later.

In order to see the role of the geometric factor, it is useful to consider the lattice version of the SSD problem, the Hamiltonian of which is defined as $$\begin{aligned}
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
{\hat {\cal H}}_{\rm lat}= -\sum_{n=1}^{L} \sin^2 \left(\frac{\pi}{L}n \right) ({\hat c}_n^\dagger {\hat c}_{n+1} + {\hat c}^\dagger_{n+1} {\hat c}_n ) 
-\mu\sum_{n=1}^{L} \sin^2 \left[ \frac{\pi}{L} \left( n-\frac{1}{2} \right) \right]{\hat c}^\dagger_n {\hat c}_n ,
\label{latticeSSD}\end{aligned}$$ where ${\hat c}^\dagger_n$ and ${\hat c}_n$ 