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 a Hamiltonian converges to one with zero-range (also called delta or point) interactions. The convergence is understood in the norm resolvent sense. The two-body rescaled potentials are of the form vσϵ(xσ)=ϵ-1vσ(ϵ-1xσ), where σ = 23, 12, 31 is an index that runs over all the possible pairings of the three particles, xσis the relative coordinate between two particles, and ϵ is the scale parameter. The limiting Hamiltonian is the one formally obtained by replacing the potentials vσ with ασδσ, where δσis the Dirac delta-distribution centered on the coincidence hyperplane xσ= 0 and ασ= Rvσdxσ. To prove the convergence of the resolvents, we make use of Faddeev's equations.