Exact quantum time of arrival correction terms for quartic potential

Abstract

The quantum correction terms of the time kernel factor are calculated analytically by converting the hyperbolic partial differenertial equation into an ordinary differential equation. It turns out that the previous correction serves as the nonhomogenous term of the ordinary differential equation satisfied by the desired correction. A sequence of function satisfying an integral recurrence relation was obtained. The results are used to further calculate the eigenvalues of the time of arrival operator.