Lieb–Liniger integral equation

Example

Consider the Lieb–Liniger model of bosons on a line with delta interaction of strength $c > 0$, in units with $\hbar = 2m = 1$. In the thermodynamic limit its ground state is described by a density $\rho$ of quasi-momenta on an interval $[-K, K]$, with $K > 0$. This density solves $$\rho(k) = \frac{1}{2\pi} + \frac{1}{\pi} \int_{-K}^{K} \frac{c}{c^2 + (k - q)^2} \, \rho(q) \, dq, \quad -K \le k \le K.$$ The particle density and the ground state energy per length are $n = \int_{-K}^{K} \rho(k) \, dk$ and $e = \int_{-K}^{K} k^2 \rho(k) \, dk$.

The equation is a Fredholm equation of the second kind on $[-K, K]$ with the continuous difference kernel $\frac{1}{\pi} \frac{c}{c^2 + (k - q)^2}$. This kernel is positive and $\frac{1}{\pi} \int_{-\infty}^{\infty} \frac{c}{c^2 + u^2} \, du = 1$. So for each $k$ the integral of the kernel over the bounded interval $[-K, K]$ is strictly less than $1$. By continuity and compactness, its maximum over $k$ is some $\theta < 1$. Hence the integral operator has norm $\theta < 1$ on $C[-K, K]$ with the supremum norm. Then the Liouville–Neumann series converges to the unique continuous solution, which is positive.