Detecting screens modeled by Schrödinger operators that generate $C_0$ contraction semigroups
Lawrence Frolov
TL;DR
The paper addresses modeling irreversible hard detection of a quantum particle confined to a bounded $C^2$ domain by detectors on the boundary. It develops a rigorous framework based on boundary quadruples to show that all such models correspond to a linear absorbing boundary condition along $\partial\Omega$, yielding a $C_0$ contraction semigroup that weakly solves the Schrödinger equation $i\partial_t\psi=\hat{H}^*\psi$ inside $\Omega$. By parameterizing these semigroups with contractions $\Phi$ between boundary data spaces, the authors derive Robin-type boundary conditions that generate well-posed dynamics and, under $\text{Re}(\beta)>0$, asymptotic detection (norm decay). Leveraging Werner's exit-space construction, they obtain a Born rule for detection times and show almost-sure finite-time detection when detectors cover the entire boundary. The results provide a comprehensive, operator-theoretic bridge between absorbing boundary conditions, dissipative extensions, and stochastic detection times for non-relativistic quantum systems. They also suggest avenues for extension to more general domains, relativistic settings, and time-dependent detector models.
Abstract
Consider a non-relativistic quantum particle with wave function $ψ$ in a bounded $C^2$ region $Ω\subset \mathbb{R}^n$, and suppose detectors are placed along the boundary $\partial Ω$. Assume the detection process is irreversible, its mechanism is time independent and also hard, i.e., detections occur only along the boundary $\partial Ω$. Under these conditions Tumulka informally argued that the dynamics of $ψ$ must be governed by a $C_0$ contraction semigroup that weakly solves the Schrödinger equation and proposed modeling the detector by a time-independent local absorbing boundary condition at $\partial Ω$. In this paper, we apply the newly discovered theory of boundary quadruples to parameterize all $C_0$ contraction semigroups whose generators extend the Schrödinger Hamiltonian, and prove a variant of Tumulka's claim: all such evolutions are generated by the placement of a linear absorbing boundary condition on $ψ$ along $\partial Ω$. 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 $\partial Ω$, and we prove that a detection will almost surely occur in finite time if detectors have been placed everywhere along $\partial Ω$.