Transparent Boundary Conditions relationship with intermediate BCs in ADI-PR method

I have read some materials about ADI - PR method with the aim to understand how to put boundary conditions in my 2D scheme which solves the Time-Dependent Schrodinger Equation.

All the theory I read starts with the assumption that the equation to be solved has Dirichlet boundary conditions of the type $$u(0, y, t) = g(y, t)$$, if the operator acting on $$u^{(n+1)*}$$ is only differencing the x-coordinate. Then they derive boundary conditions for the intermediate timestep function, and they arrive at an expression similar to $$U^*_{0, m} = ... = coeff_1 * g_{m}^{n} + coeff_2 * g_{m}^{n+1}$$, where $$g_m^{n+1} = g(mh, t_n + k)$$, where $$h$$ is the cell length in both $$x$$ and $$y$$ directions and $$k$$ is the timestep duration. The $$U^*_{0, m}$$ approximates the solution to the first equation from the set of 2 from the ADI - PR method. Thus they assume one has $$g$$ in closed form and can query it at any timestep $$t$$ and at any $$y$$.

My problem requires the implementation of Transparent Boundary Conditions which read, for a 1D problem, $$u^n_{0} = \alpha u^{n}_1$$, where $$\alpha = u^{n-1}_1 / u^{n-1}_2$$. Thus the boundary condition is dependent on the previous iterations. I do not have that $$g$$ from mathematics literature, and I cannot query it at timestep $$t_n + k$$.

How can I go about implementing these transparent boundary conditions in my 2D ADI - PR solver?

Thank you!