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 $v_{\sigma }^{\epsilon }(x_{\sigma })={\epsilon }^{-1}{v}_{\sigma }({\epsilon }^{-1}{x}_{\sigma })$, 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 $v_{\sigma }$ with ${\alpha }_{\sigma }{\delta }_{\sigma }$, where ${\delta }_{\sigma }$ is the Dirac delta-distribution centered on the coincidence hyperplane $x_{\sigma }=0$ and ${\alpha }_{\sigma }={\int }_{\mathbb{R}}{v}_{\sigma }d{x}_{\sigma }$. To prove the convergence of the resolvents we make use of Faddeev's equations.

arXiv comment21 pages

References in corpus (2)

• On point interactions realised as Ter-Martirosyan-Skornyakov Hamiltonians
• N-body problem with contact interactions

Cited by in corpus (8)

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• Energy contribution of a point interacting impurity in a Fermi gas
• On the Weakness of Short-Range Interactions in Fermi Gases