标题： Schrödinger 算子的延拓生成 $C_0$ 契约半群 显示英文标题

标题： Extensions of Schrödinger operators that generate $C_0$ contraction semigroups

摘要： 考虑一个非相对论量子粒子，其波函数为$\psi$，处于有界区域$C^2$内的$\Omega \subset \mathbb{R}^n$区域，假设在边界$\partial \Omega$处放置了探测器。 假定探测过程是不可逆的，其机制与时间无关且非常“硬”，即探测仅发生在边界$\partial \Omega$上。 在此条件下，图穆尔卡论证了 $\psi$ 的动力学必须由一个 $C_0$ 型的收缩半群支配，该半群弱解薛定谔方程，并提出用时间无关的局部吸收边界条件来模拟探测器于 $\partial \Omega$ 处。本文中，我们应用新发现的边界四元组理论来参数化所有扩展了薛定谔哈密顿算符的生成元的 $C_0$ 型收缩半群，并证明了图穆尔卡主张的一个变体：所有此类演化均由在 $\psi$ 上放置（可能非局域的）吸收边界条件于 $\partial \Omega$ 上生成。 我们将这一结果与Werner的工作相结合，证明每个$C_0$压缩半群自然地允许沿$\partial \Omega$的检测时间的概率分布，并且我们证明了对于一大类吸收边界条件，粒子被检测到的概率等于$1$。 显示英文摘要

摘要： 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 $1$.