Existence of Schrödinger 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 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.