quantum mechanics - The $iepsilon$ in non-relativistic scattering theory

When doing quantum mechanical scattering theory, we obtain the Lippman-Schwinger equation

$$|\psi\rangle=|\psi_0\rangle+(E-H_0)^{-1}V|\psi\rangle$$

Here $\psi_0$ is the unperturbed wavefunction, $\psi$ is the total wavefunction, and we define $\psi_s=\psi_0-\psi$ is the scattered wave function. When solving this, people usually replace $(E-H_0)\rightarrow E-H_0+i\epsilon$ while mumbling something about causality and $\psi_s$ not having incoming probability current. Then of course, in the final result, we indeed see that we obtain something reasonable.

However, I would like to know if there is some more robust and mathematical way to derive the $i\epsilon$ part, instead of just putting it in by hand and see that the final result is reasonable? For example, if one could start with the initial Schrodinger equation, and demand no incoming probabiltiy current on the scattered wave to see that the $i\epsilon$ comes out naturally?