Existence of Schrodinger Evolution with Absorbing Boundary Condition

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 \emph{absorbing boundary rule}, involves a time evolution for the particle's wave function $\psi$ expressed by a Schrödinger 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, under some technical assumptions on the regularity (i.e., smoothness) of the detecting surface, the Lumer-Phillips 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. We also prove that the joint distribution of the detection times and places, according to the absorbing boundary rule, is governed by a positive-operator-valued measure.

21 pages LaTeX, no figures; v5 minor revision; in v4, an error in Theorem 1 has been corrected, and the treatment of the Dirac case postponed to a future paper

References in corpus (8)

• Quantum Dynamics under continuous projective measurements: non-Hermitian description and the continuous space limit
• Quantum Hamiltonians and Stochastic Jumps
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• Arrival Times Versus Detection Times
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Cited by in corpus (7)

• Quantum Dynamics under continuous projective measurements: non-Hermitian description and the continuous space limit
• Absorbing Boundary Condition as Limiting Case of Imaginary Potentials
• Detecting screens modeled by Schrödinger operators that generate $C_0$ contraction semigroups
• Absorbing detectors meet scattering theory
• A solution of the quantum time of arrival problem via mathematical probability theory
• Detection Time Distribution Predicted Using Absorbing Boundary Conditions and Imaginary Potentials
• Energy-Time Uncertainty Relation for Absorbing Boundaries