arXiv:1204.0435 · Journal of Spectral Theory

Absence of bound states implies non-negativity of the scattering length

Robert Seiringer

Published Jun 24, 2012• 5 citations

Abstract

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.

DOI: 10.4171/jst/31

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arXiv:1204.0435 · Journal of Spectral Theory

Absence of bound states implies non-negativity of the scattering length

Robert Seiringer

Published Jun 24, 2012• 5 citations

arXiv HTML PDF

Abstract

DOI: 10.4171/jst/31

Related papers

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In this third paper of a series that started with arXiv:2106.10032 [math-ph] and continued with arXiv:2108.02659 [math-ph] we show that in

d\geq 3

dimensions at low temperatures or high densities bosons interacting via pair potentials that are both positive and positive type form permutation cycles whose length diverges proportionally with the number of particles. Based on the second-cited paper, this implies Bose-Einstein condensation.

We consider bosons interacting through a narrow

s

-wave resonance. Such a resonance is characterized by an infinite scattering length and a large and negative effective range

r_0

. We argue that any number

N\ge3

of bosons can form a self-bound cluster with the binding energy per particle increasing as

N^2

for

1\ll N\ll (-r_0/a_\text{bg})^{1/2}

, where

a_\text{bg}

is the background scattering length (between atoms and molecules). In the opposite limit

N\gg (-r_0/a_\text{bg})^{1/2}

, bosons form droplets with binding energy per particle saturating to a constant value independent of the particle number. The stability of clusters and droplets when the interaction is detuned from the resonance is also studied.