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

Abstract

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 $v^{\varepsilon}_σ(x_σ)= \varepsilon^{-1} v_σ(\varepsilon^{-1}x_σ)$, where $σ= 23, 12, 31$ is an index that runs over all the possible pairings of the three particles, $x_σ$ 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 $v_σ$ with $α_σδ_σ$, where $δ_σ$ is the Dirac delta-distribution centered on the coincidence hyperplane $x_σ=0$ and $α_σ= \int_{\mathbb{R}} v_σdx_σ$. To prove the convergence of the resolvents we make use of Faddeev's equations.

21 pages