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The interactions between atoms and molecules with each other and with surfaces are typically deep potential wells with attractive tails behaving asymptotically as an inverse power of the distance. In such potential wells, bound state energies $E_n$ are determined by a quantization function $F\left(E\right)$ according to $n_{\text{th}}-n=F\left(E_n\right)$, and $F\left(E\right)$ is dominantly determined by the singular potential tail for near-threshold states. We present general expressions for the contribution $F_{\text{tail}}$ of the singular potential tail to the quantization function for near-threshold energies in two and three dimensions. For homogeneous potential tails proportional to $-1/r^{\alpha}$, analytical formulas for $F_{\text{tail}}$ are given and we show the connection between the scattering length $a$ and near-threshold energies $E_n$. In three dimensions we furthermore present an interpolated formula for the tail contribution for arbitrary energies and we demonstrate how the dissociation energy of a diatomic molecule can be determined from spectroscopic energies of high-lying states. |