Universality in a one-dimensional three-body system

Abstract

We study a heavy-heavy-light three-body system confined to one space dimension. Both binding energies and corresponding wave functions are obtained for (i) the zero-range, and (ii) two finiterange attractive heavy-light interaction potentials. In case of the zero-range potential, we apply the method of Skorniakov and Ter-Martirosian to explore the accuracy of the Born-Oppenheimer approach. For the finite-range potentials, we solve the Schrödinger equation numerically using a pseudospectral method. We demonstrate that when the two-body ground state energy approaches zero, the three-body bound states display a universal behavior, independent of the shape of the interaction potential.

Tensor Product Scheme for Computing Bound States of the Quantum Mechanical Three-Body Problem

“…Moreover, lower dimensional systems, such as three fermionic particles confined to two dimensions, display surprising universal phenomena like the so-called "super Efimov effect" [3,4,5,6]. Recently, some of the authors have demonstrated that in one dimension, universality in a mass-imbalanced three-body system is not only present in the discrete spectrum [7], but universal three-body bound states also occur in the continuum as induced by excited two-body resonances [8].…”

“…For instance, in Ref. [7] three-body energies and the corresponding wave functions of the bound states are computed by representing the Hamiltonian as a sparse matrix, and using a Krylov subspace method to determine its lowest eigenvalues and the corresponding eigenvectors. However, with an increasing number of grid points in each dimension, the matrices and vectors grow rapidly: The three-body problem in d space dimensions yields a 2d-dimensional linear eigenvalue problem after removing the center-of-mass degree of freedom.…”

“…When discretized with n grid points in each direction, a single vector that represents the three-body wave function has the size n 2d . Moreover, the pseudo-spectral discretization used in [7] leads to O n 2d−1 dense blocks in the sparse matrix representation of the Hamiltonian, each of size n × n. Thus, such an approach is severely limited by the 'curse of dimensionality'.…”

Direct Measurements of Collisional Dynamics in Cold Atom Triads

“…Three-body recombination occurs throughout physics from ultracold plasmas [33] to chemistry [34] and astrophysics [35] and has been extensively studied in ultracold atoms [22,[36][37][38][39][40][41]. Moreover, the rich physics of idealised three atom systems in tightly confining traps is currently the target of intensive theoretical studies [42][43][44][45][46][47], while experiments are presently lacking.…”

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

“…and remains finite also in the limit q 0 → 0, due to equation (12). First, we discuss the integrals I 1 and I 4 .…”