Abstract
We provide an analytical proof of universality for bound states in one-dimensional systems of two and three particles, valid for short-range interactions with negative or vanishing integral over space. The proof is performed in the limit of weak pair-interactions and covers both binding energies and wave functions. Moreover, in this limit the results are formally shown to converge to the respective ones found in the case of the zero-range contact interaction.
Highlights
In contrast to purely attractive potentials, which are ubiquitous in quantum physics, interactions whose attractive and repulsive parts cancel each other are only scarcely discussed
The latter allow for bound states [1], and interest in such potentials ramped up within the last years with the ability to realize them in systems of ultracold dipoles [2]
This is supported by recent analysis for these potentials on the formation of few- and many-body bound states in arrays of one-dimensional tubes [3] or in terms of beyond-mean-field contributions in reduced dimensions [4]
Summary
In contrast to purely attractive potentials, which are ubiquitous in quantum physics, interactions whose attractive and repulsive parts cancel each other are only scarcely discussed The latter allow for bound states [1], and interest in such potentials ramped up within the last years with the ability to realize them in systems of ultracold dipoles [2]. We consider only short-range interactions v(ξ) ≡ v0 f (ξ) with magnitude v0 and shape f (ξ) between distinguishable particles, and none between identical ones Within these systems we are interested in the universal behavior [9, 10] and consider the weaklyinteracting limit v0 → 0, which implies [1, 11] a weakly-bound two-body ground state. In order to simplify the analysis we introduce the scaled momentum P ≡ p/q0 with q0 ≡ 2 E0(2)
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