Isotropic contact forces in arbitrary representation: Heterogeneous few-body problems and low dimensions

The Bethe-Peierls asymptotic approach which models pairwise short-range forces by contact conditions is introduced in arbitrary representation for spatial dimensions less than or equal to 3. The formalism is applied in various situations and emphasis is put on the momentum representation. In the presence of a transverse harmonic confinement, dimensional reduction toward two-dimensional (2D) or one-dimensional (1D) physics is derived within this formalism. The energy theorem relating the mean energy of an interacting system to the asymptotic behavior of the one-particle density matrix illustrates the method in its second quantized form. Integral equations that encapsulate the Bethe-Peierls contact condition for few-body systems are derived. In three dimensions, for three-body systems supporting Efimov states, a nodal condition is introduced in order to obtain universal results from the Skorniakov–Ter-Martirosian equation and the Thomas collapse is avoided. Four-body bound state eigenequations are derived and the 2D ${}^{\prime }3+1^{\prime }$ bosonic ground state is computed as a function of the mass ratio.

Figure 1

Shape of the expession of $k^2\varphi (k)$ in Eq. (118) plotted in a semi-log plot as a function of $k/q$.

Figure 2

Solid line: Second branch of the trimer’s spectrum obtained with the nodal condition at $k_2^{\mathrm{reg}}$. Vertical dashed line: Trimer’s appearance thresholds. Oblique dashed line: Atom-dimer continuum limit. The values of the scattering length at the trimer’s appearance threshold, $a^{\star }\simeq -1.51/{\kappa }^{\star }$ and a’${}^{*}\simeq .0713/{\kappa }^{\star }$, are close to the results of the “universal theory” in Refs. [35, 62], ${\mathrm{a}}^{*}\simeq -1.507/{\kappa }^{\star }$ and a’${}^{*}\simeq .0707/{\kappa }^{\star }$. The difference is due to the fact that the zero-range theory is recovered only for a nodal condition at an arbitrarily large $k_p^{\mathrm{reg}}$.

Figure 3

Atom-dimer scattering length $a_{\mathrm{ad}}$ computed with the nodal condition at $k_2^{\mathrm{reg}}$ and plotted in a semi-log scale as a function of ${\kappa }^{\star }/a_3$ for $a_3{k}_2^{\mathrm{reg}}\ll 1$.

Figure 4

Schematic representation of the momentum coordinates used for the source amplitude associated with the contact condition between particles 1 and 3.

Figure 5

Critical mass ratio $\left(\frac{M}{m}\right){}_l^{\mathrm{crit}}$ for the threshold of appearance of an Efimov spectrum in each partial wave $l$ of Eq. (137).

Figure 6

Schematic representation of the momentum coordinates used for the source amplitude associated with the contact condition between particles 1 and 2 for a system of fermions with two-mass components.

Figure 7

Ground-state branch for the three bosons of mass $M$ interacting with one impurity of mass $m$ in 2D as a function of the mass ratio.