高斯势的$s$波散射长度
The $s$-wave scattering length of a Gaussian potential
摘要：
我们提供了精确的表达式的$s$波散射长度的高斯势阱在一个，两个和三个空间维度。高斯势作为赝势被广泛应用于超冷原子气体的理论描述中，其中$s$波散射长度是一个物理相关参数。我们首先描述了一个数值计算过程的$s$ -波散射长度的高斯参数的值，但发现它的准确性是有限的，在附近的奇点，导致新的束缚态的形成。然后，我们推导出简单的解析表达式，捕捉正确的渐近行为的$s$波散射长度附近的束缚态。越来越准确的表达式在宽参数制度发现的层次结构的近似，捕获越来越多的束缚态。进入这些表达式的少量数值系数由精确的数值计算确定。近似公式结合了数值表达式和近似表达式的优点，给出了从弱相互作用极限到强相互作用极限的准确而简单的描述。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
