consider a one-dimensional gas of $N$ neutral bosons of mass $m$ which are subject to an external potential $V_\mathrm{ext}(x_n)$. The Hamiltonian is given by $$H= \sum^N_{n=1} \left[ -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x_n^2}+V_\mathrm{ext}(x_n) \right]+ g \sum_{i<j}\delta(|x_i-x_j|),
\label{eq:basicmodel}$$ where the two-body interaction between the particles is described by a zero-range delta-function potential of strength $g$ [@Olshanii:98]. While exact solutions for this Hamiltonian exist in the absence of an external potential [@Sutherland:04; @Franchini:17], only limiting cases can be solved in other situations.

One regime in which the Hamiltonian can be solved exactly is the strongly interacting Tonks-Girardeau limit. It corresponds to setting $g=\infty$, which allows one to obtain an exact solution for arbitrary $N$ by utilizing the Bose-Fermi mapping theorem [@Girardeau1960]. The basic idea behind this theorem is to replace the interaction term by a constraint on the many-body wavefunction of the form $$\Psi(x_1,...x_N) =0 \qquad  \mathrm{if} \qquad |x_i-x_j|=0,$$ which is equivalent to enforcing the Pauli exclusion principle in position space. The strongly repulsive bosonic system can therefore be mapped onto a gas of non-interacting, spinless fermions for which the many-body wave-function can be written as a sum of single particle product states using the Slater determinant $$\Psi_F(x_1,...,x_N)=\frac{1}{\sqrt{N!}} \mathrm{det}^{(N,N)}_{(n,i)=(1,1)}\psi_n(x_i).$$ Here the $\psi_n(x)$ are the eigenfunctions of 