parameter $\mu$, one can obtain them in closed form by exploiting the hidden SUSY in the problem.

Single-particle eigenstates of the SSD problem {#sec:susy_solution}
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In this section, we present a systematic procedure to construct single-particle eigenstates of the SSD problem. To this end, we introduce the following parametrization of $\xi$: $$\xi = - \left(\frac{\pi}{L}\right)^2\beta (\beta-1),
\label{SSDspectrum}$$ and rewrite Eq. (\[ssdh\]) with $f(x)=\sin^2 \frac{\pi}{L}x$ as $$H_\beta\, \varphi (x) = \mu \varphi (x),
\label{SSDpotential}$$ where the “Hamiltonian" $H_\beta$ is defined as $$H_\beta = - \frac{d^2}{dx^2} + \left(\frac{\pi}{L}\right)^2 
\frac{\beta (\beta-1)}{\sin^2 \frac{\pi}{L} x }.$$ In the standard setup, “energy eigenvalue" $\mu$ is determined for a given parameter $\xi$ (or, equivalently, $\beta$), where the corresponding eigenfunction has a wavepacket-like shape. On the other hand, however, what we need here is to determine the energy eigenvalue $\xi$ for a given chemical potential $\mu$. In this sense, the SSD problem can be formulated as the inverse problem of the $1/\sin^2$ potential problem.

It is known that the $1/\sin^2$ potential problem can be solved algebraically since the system has SUSY and shape invariance [@SUSYQM]. This property is also deeply related to the integrable many-body system with an inverse sine-square interaction solved by Sutherland [@Sutherland; @Kuramoto]. Here, we 