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 upsilon(epsilon)(sigma) (x(sigma)) = epsilon(-1) v(sigma) (epsilon(-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 epsilon is the scale parameter. The limiting Hamiltonian is the one formally obtained by replacing the potentials upsilon(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) = integral(R) upsilon(sigma)dx(sigma). To prove the convergence of the resolvents, we make use of Faddeev's equations. Published by AIP Publishing.