Detecting screens modeled by Schr\"odinger operators that generate $C_0$ contraction semigroups

Consider a non-relativistic quantum particle with wave function $\psi$ in a bounded $C^2$ region $\Omega \subset \mathbb{R}^n$, and suppose detectors are placed along the boundary $\partial \Omega$. Assume the detection process is irreversible, its mechanism is time independent and also hard, i.e., detections occur only along the boundary $\partial \Omega$. Under these conditions Tumulka informally argued that the dynamics of $\psi$ must be governed by a $C_0$ contraction semigroup that weakly solves the Schr\"odinger equation and proposed modeling the detector by a time-independent local absorbing boundary condition at $\partial \Omega$. In this paper, we apply the newly discovered theory of boundary quadruples to parameterize all $C_0$ contraction semigroups whose generators extend the Schr\"odinger Hamiltonian, and prove a variant of Tumulka's claim: all such evolutions are generated by the placement of a linear absorbing boundary condition on $\psi$ along $\partial \Omega$. We combine this result with the work of Werner to show that each $C_0$ contraction semigroup naturally admits a Born rule for the time of detection along $\partial \Omega$, and we prove that a detection will almost surely occur in finite time if detectors have been placed everywhere along $\partial \Omega$.