Scattering and Bound States

Lanie G. McKinney, Brett Allen McKinney

Abstract We determine the conditions for bound states (E < 0) for arbitrary Cartesian dimension d using a shape-independent regularized pseudopotential with scattering length a for two cold particles in a harmonic trap. It is known for d ≤ 3 that the regularized pseudopotential supports one bound state for positive scattering length but does not support bound states for negative scattering length. We find that the usual (d ≤ 3) positive scattering length bound states rule holds for certain higher odd dimensions d = 4n + 3 (n = 0, 1, …), but the existence of pseudopotential bound states at other odd dimensions requires a negative scattering length. Specifically, bound states are allowed in higher dimensions d = 4n + 1 (n = 1, 2, …) but they require a negative scattering length, which suggests a universe in these dimensions might lead to different chemistry than d = 3. We derive analytical approximations for bound state (E < 0) and scattering (E > 0) energies for a harmonic trap perturbed by the pseudopotential in arbitrary dimensions.

Stephen J. Gustafson, Israel Michael Sigal

Stephen J. Gustafson, Israel Michael Sigal

T. K. Lim, K E Duffy

Faddeev–UPE theory is applied to the trimer bound states and atom–diatom scattering of 4He. (4He)3 has two bound states while the scattering length of 4He– (4He)2 is 207 Å.

Stephen J. Gustafson, Israel Michael Sigal

Robert Seiringer

We show that bosons interacting via pair potentials with negative scattering length form bound states for a suitable number of particles. In other words, the absence of many-particle bound states of any kind implies the non-negativity of the scattering length of the interaction potential.