We investigate how the left-hand cut (LHC) problem is treated in the HAL QCD method.
For this purpose, we first consider the effect of the LHC to the scattering problem in non-relativistic quantum mechanics with potentials.
We show that the $S$-matrix or the scattering phase shift obtained from the potential including the Yukawa term ($e^{- m_\pi r}/r$) with the infra-red (IR) cutoff $R$ is well-defined even for the complex momentum $k$ as long as $R$ is finite, and they are compared with those obtained by the analytic continuation without the IR cutoff.
In the $R\to\infty$ limit, the phase shift approaches the result from the analytic continuation at ${\rm Im}\, k < m_\pi/2$, while they differ at ${\rm Im}\, k > m_\pi/2$, except $k= k_b$, where $k_b$ is the binding momentum.
We also observe that $k_b$ can be correctly obtained even at finite but large $R$.
Using knowledge obtained in the non-relativistic quantum mechanics, we present how we should treat the LHC in the HAL QCD potential method.