Boundary conditions for time-dependent quantum slits

We prove that for time-dependent quantum slits, the widely used Kirchhoff boundary condition (BC) is an approximation valid only for long times and so predicts unreliable transmitted wave functions at the transient regime. We derive, for arbitrary time-dependent slits, an integral equation for the general theory of the boundary function. We derive a new boundary function valid for hight opening velocities, which for short times has the correct quantum behavior, and for long times we recover the Kirchhoff BC as a particular case