Existence of Schrodinger Evolution with Absorbing Boundary Condition

• URL: http://arxiv.org/abs/1912.12057v2
• Date: Fri, 29 Jul 2022 11:55:17 GMT
• Title: Existence of Schrodinger Evolution with Absorbing Boundary Condition
• Authors: Stefan Teufel, Roderich Tumulka
• Abstract summary: Consider a non-relativistic quantum particle with wave function inside a region $Omegasubset mathbbR3$. The question how to compute the probability distribution of the time at which the detector surface registers the particle boils down to finding a reasonable mathematical definition of an ideal detecting surface. A particularly convincing definition, called the absorbing boundary rule, involves a time evolution for the particle's wave function $psi$ expressed by a Schrodinger equation in $Omega$ together with an "absorbing" boundary condition on $partial Omega$ first considered by Werner in
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Consider a non-relativistic quantum particle with wave function inside a region $\Omega\subset \mathbb{R}^3$, and suppose that detectors are placed along the boundary $\partial \Omega$. The question how to compute the probability distribution of the time at which the detector surface registers the particle boils down to finding a reasonable mathematical definition of an ideal detecting surface; a particularly convincing definition, called the absorbing boundary rule, involves a time evolution for the particle's wave function $\psi$ expressed by a Schrodinger equation in $\Omega$ together with an "absorbing" boundary condition on $\partial \Omega$ first considered by Werner in 1987, viz., $\partial \psi/\partial n=i\kappa\psi$ with $\kappa>0$ and $\partial/\partial n$ the normal derivative. We provide here a discussion of the rigorous mathematical foundation of this rule. First, for the viability of the rule it plays a crucial role that these two equations together uniquely define the time evolution of $\psi$; we point out here how the Hille-Yosida theorem implies that the time evolution is well defined and given by a contraction semigroup. Second, we show that the collapse required for the $N$-particle version of the problem is well defined. Finally, we also prove analogous results for the Dirac equation.

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