Construction and dynamics of algebra-preserving time of arrival operators

Abstract

The construction of algebra-preserving time of arrival (APTOA) operators for nonlinear potentials has been a longstanding challenge in the theory of supraquantization. In this paper, we construct such an operator using a sinusoidal potential. The construction is facilitated by solving the associated time kernel equation (TKE) in closed-form using a specific separability condition. We then demonstrate numerically that the APTOA operator's eigenfunctions have the desired unitary arrival property, linking TOA measurements and the appearance of the incident particle at the arrival point. Finally, we show that the same operator exhibits sharper unitary dynamics compared to a non-algebra-preserving operator derived from Weyl quantization of the classical arrival time.