Extensions of Schr\"odinger operators that generate C₀ contraction semigroups

Consider a non-relativistic quantum particle with wave function ψ in a bounded C² region Ω ⊂ Rⁿ, and suppose detectors are placed along the boundary ∂ Ω. Assume the detection process is irreversible, its mechanism is time independent and also hard, i.e., detections occur only along the boundary ∂ Ω. Under these conditions Tumulka argued that the dynamics of ψ must be governed by a C₀ contraction semigroup that weakly solves the Schr\"odinger equation and proposed modeling the detector by a time-independent local absorbing boundary condition at ∂ Ω. In this paper, we apply the newly discovered theory of boundary quadruples to parameterize all C₀ 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 (potentially nonlocal) absorbing boundary conditions on ψ along ∂ Ω. We combine this result with the work of Werner to show that each C₀ contraction semigroup naturally admits a probability distribution for the time of detection along ∂ Ω, and we prove for a wide class of absorbing boundary conditions that the probability of the particle being ever detected is equal to $1$.