  The construction of time of arrival (TOA) operators canonically conjugate to
the system Hamiltonian entails finding the solution of a specific second-order
partial differential equation called the time kernel equation (TKE). In this
paper, we provide an exact analytic solution of the TKE for a special class of
potentials satisfying a specific separability condition. The solution enables
us to investigate the time evolution of the eigenfunctions of the
conjugacy-preserving TOA operators (CPTOA) and show that they exhibit unitary
arrival at the intended arrival point at a time equal to their corresponding
eigenvalues. We also compare the dynamics between the TOA operators constructed
by quantization and those independent of quantization for specific interaction
potentials. We find that the CPTOA operator possesses smoother and sharper
unitary dynamics over the Weyl-quantized one within numerical accuracy.
