I’m interested in solving the 1D time-dependent Schrodinger equation $ i partial_t u(x,t) = -partial_x^2 u(x,t) + V(x,t) u(x,t) $ with a transparent boundary condition at, for instance, $x = x_0$ :

$$

partial_x u(x_0,t) propto e^{-i V t} frac{d}{dt} int_0^t frac{u(x_0,tau) e^{i V tau}}{sqrt{t-tau}} d tau

$$

which is non-local in the variable $t$.

I was not able to implement that boundary condition “as is” into `NDSolve()`

…

Is there a way to use that non-local boundary condition with `NDSolve()`

?

Thanks in advance.