Shifted solution to the time kernel equation

Abstract

The solution to the time kernel equation with the boundary conditions T(q,q) = q/2 and T(q,−q) = 0 correspond to the quantum time of arrival 𝒯_TOA ∝ T(q,q'). Here, we investigate a different set of conditions, i.e., T(q,q) = q/2 and T(q,−q) = −2iβq, for some constant β. We show that these conditions correspond to the shifted solution 𝒯_TOA + βℏH⁻¹, where H is the Hamiltonian.