Classical aspects of quantum walls in one dimension

We investigate the system of a particle moving on a half line $x>~0$ under the general walls at $x=0\mathrm{}$ that are permitted quantum mechanically. These quantum walls, characterized by a parameter L, are shown to be realized as a limit of regularized potentials. We then study the classical aspects of the quantum walls by seeking a classical counterpart that admits the same time delay in scattering with the quantum wall, and also by examining the WKB exactness of the transition kernel based on the regularized potentials. It is shown that no classical counterpart exists for walls with $L<\mathrm{}0,$ and that the WKB exactness can hold only for $L=0\mathrm{}$ and $L=\infty .$