The three-body problem in dimension one: From short-range to contact interactions

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.