General form of separable potentials leading to exact closed-form solution of the time kernel equation

Abstract

In investigating time-of-arrival observables (TOA) that are canonically conjugate to the Hamiltonian, an exact closed-form expression will be of inevitable necessity. We have previously derived such expression but is only applicable for a class of potentials we refer to as separable potentials. In this paper, we derive the necessary form separable potentials must assume. We found that they must be a superposition of linearly independent exponential functions and some constant. This provides an explicit closed-form formula valid for potentials with said form.