Proof of universality in one-dimensional few-body systems including anisotropic interactions

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.

Universality of excited three-body bound states in one dimension

“…In this section we present the energies and wave functions of three-body bound states associated with the two-body threshold E (2) r → 0 − for r = 1,r = 2 and r = 3. For this purpose, we have solved equation (18) numerically for two finiterange interaction potentials of shape f L , equation (3), and f G , equation (4). By varying the magnitude v 0 of the potential, we can tune the heavy-light subsystems to a particular threshold.…”

“…In figure 5 we illustrate the influence of the deeply-bound states on the energy ratio ˜ r,n . This ratio converges towards r,n , equation (9), provided that all expansion terms are included in equations (15) and (18). To explore the influence of particular expansion terms on the three-body energy, we consider different combinations in the expansion as indicated by the legend.…”

“…In this appendix we derive the integral equation, equation (18), corresponding to the three-body Schrödinger equation, equation (7) within the Faddeev approach [20,58] and the separable expansion [19][20][21]. We consider the three-body system as introduced in section 2.2, where the light particle of mass m is called particle 1, and particle 2 and 3 are the two identical heavy ones of mass M.…”

“…Recently, we have proven universality [17,18] in a heavy-heavy-light three-body system, which is confined to one dimension (1D). The universality requires the heavy-light interactions to be tuned towards the ground-state threshold, that is the binding energy of the ground state approaches zero.…”