Related papers: Extensions of Schrödinger operators that generate $C

Extensions of Schrödinger operators that generate $C_0$ contraction semigroups

URL: http://arxiv.org/abs/2506.00231v1
Date: Fri, 30 May 2025 21:08:15 GMT
Title: Extensions of Schrödinger operators that generate $C_0$ contraction semigroups
Authors: Lawrence Frolov,
Abstract summary: Tumulka argued that the dynamics of $psi$ must be governed by a $C_0$ contraction semigroup.<n>We show that all such evolutions are generated by the placement of (potentially nonlocal) absorbing boundary conditions on $psi$ along $partial Omega$.
Score: 0.0
License: http://creativecommons.org/licenses/by/4.0/
Abstract: 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 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 (potentially nonlocal) absorbing boundary conditions 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 probability distribution for the time of detection along $\partial \Omega$, and we prove for a wide class of absorbing boundary conditions that the probability of the particle being ever detected is equal to $1$.

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