Detecting screens modeled by Schrödinger operators that generate $C_0$ contraction semigroups
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911727576154112 |
|---|---|
| author | Frolov, Lawrence |
| author_facet | Frolov, Lawrence |
| contents | 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 $\partial Ω$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_00231 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Detecting screens modeled by Schrödinger operators that generate $C_0$ contraction semigroups Frolov, Lawrence Mathematical Physics Quantum Physics 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 $\partial Ω$. |
| title | Detecting screens modeled by Schrödinger operators that generate $C_0$ contraction semigroups |
| topic | Mathematical Physics Quantum Physics |
| url | https://arxiv.org/abs/2506.00231 |