  We consider a Hamiltonian describing three quantum particles in dimension one
interacting through two-body short-range potentials. We prove that, as a
suitable scale parameter in the potential terms goes to zero, such Hamiltonian
converges to one with zero-range (also called delta or point) interactions. The
convergence is understood in norm resolvent sense. The two-body rescaled
potentials are of the form $v^{\varepsilon}_{\sigma}(x_{\sigma})=
\varepsilon^{-1} v_{\sigma}(\varepsilon^{-1}x_\sigma )$, where $\sigma = 23,
12, 31$ is an index that runs over all the possible pairings of the three
particles, $x_{\sigma}$ is the relative coordinate between two particles, and
$\varepsilon$ is the scale parameter. The limiting Hamiltonian is the one
formally obtained by replacing the potentials $v_\sigma$ with $\alpha_\sigma
\delta_\sigma$, where $\delta_\sigma$ is the Dirac delta-distribution centered
on the coincidence hyperplane $x_\sigma=0$ and $\alpha_\sigma =
\int_{\mathbb{R}} v_\sigma dx_\sigma$. To prove the convergence of the
resolvents we make use of Faddeev's equations.
