$s$-wave scattering length of a Gaussian potential

We provide accurate expressions for the $s$-wave scattering length for a Gaussian potential well in one, two, and three spatial dimensions. The Gaussian potential is widely used as a pseudopotential in the theoretical description of ultracold-atomic gases, where the $s$-wave scattering length is a physically relevant parameter. We first describe a numerical procedure to compute the value of the $s$-wave scattering length from the parameters of the Gaussian, but find that its accuracy is limited in the vicinity of singularities that result from the formation of new bound states. We then derive simple analytical expressions that capture the correct asymptotic behavior of the $s$-wave scattering length near the bound states. Expressions that are increasingly accurate in wide parameter regimes are found by a hierarchy of approximations that capture an increasing number of bound states. The small number of numerical coefficients that enter these expressions is determined from accurate numerical calculations. The approximate formulas combine the advantages of the numerical and approximate expressions, yielding an accurate and simple description from the weakly to the strongly interacting limit.

Figure 1

Relative error in the two-dimensional $s$-wave scattering length compared to a reference value $a_s^{2D}$ computed with parameters $p=11, r=10L, \epsilon ={10}^{-6}L$. Shown is the parameter dependence (a) on the numerical precision $p$, (b) on the cutoff distance $r$, and (c) on the boundary parameter $\epsilon$ for different values of the potential strength $V_0$.

Figure 2

Relative error in the three-dimensional $s$-wave scattering length compared to a reference value $a_s^{3D}$ computed with parameters $p=11, r=10L$. Shown is the parameter dependence (a) on the numerical precision $p$ and (b) on the cutoff distance $r$ for different values of the potential strength $V_0$.

Figure 3

(a) Three-dimensional $s$-wave scattering length from the numerical and the approximate expression. (b) The difference between the approximative and numerical scattering length values. The parameters of the numerical simulations are set to $p=11$ and $r=10L$. The values of the parameters for the approximative expressions can be found in Table 1.

Figure 4

(a) One-dimensional $s$-wave scattering length from the numerical and the approximate expression. (b) The difference between the approximate and numerical values of the scattering length. The parameters of the numerical simulations are set to $p=11, r=10L$. The parameter values for the approximate expressions are tabulated in Table 2. The $n=0$ approximation corresponds to Eq. (29).

Figure 5

(a) Two-dimensional $s$-wave scattering length from the numerical and the approximate expression. (b) The difference between the approximative and numerical scattering length values. The parameters of the numerical simulations are set to the following: $p=11, r=10L$, and $\epsilon ={10}^{-6}L$. The values of the parameters for the approximative expressions can be found in Table 3. The $n=0$ approximations correspond to Eq. (35).

Figure 6

Importance of the location of the first pole on the accuracy of the scattering length in three dimensions. (a) $s$-wave scattering length from the numerical and the approximate expressions. In the approximate expressions, the scattering length is calculated with varying precision (ndigit decimal digits) of the position of the first singularity $W_1$. (b),(c) The errors of the approximate and numerical values of the scattering length. (b) $V_0<W_1$, (c) $V_0>W_1$. Reference values for determining the zero points of the $x$ and $y$ axes, respectively, are calculated by numerical calculations with a high level of accuracy ($p=15$ for $a_s$; reference value $W_1=2.684004650924{\hslash }^2/\mu$).