Existence of Schrödinger Evolution with Absorbing Boundary Condition

Abstract

Abstract Consider a non-relativistic quantum particle with wave function inside a region $$\Omega \subset \mathbb {R}^3$$ Ω ⊂ R 3 , and suppose that detectors are placed along the boundary $$\partial \Omega $$ ∂ Ω . 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 $$\psi $$ ψ expressed by a Schrödinger equation in $$\Omega $$ Ω together with an “absorbing” boundary condition on $$\partial \Omega $$ ∂ Ω first considered by Werner in 1987, viz., $$\partial \psi /\partial n=i\kappa \psi $$ ∂ ψ / ∂ n = i κ ψ with $$\kappa >0$$ κ > 0 and $$\partial /\partial n$$ ∂ / ∂ 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 $$\psi $$ ψ ; 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.