Absorbing Boundary Condition as Limiting Case of Imaginary Potentials

Imaginary potentials such as $V(x)=-iv 1_\Omega(x)$ (with $v>0$ a constant, $\Omega$ a subset of 3-space, and $1_\Omega$ its characteristic function) have been used in quantum mechanics as models of a detector. They represent the effect of a "soft" detector that takes a while to notice a particle in the detector volume $\Omega$. In order to model a "hard" detector (i.e., one that registers a particle as soon as it enters $\Omega$), one may think of taking the limit $v\to\infty$ of increasing detector strength $v$. However, as pointed out by Allcock, in this limit the particle never enters $\Omega$; its wave function gets reflected at the boundary $\partial \Omega$ of $\Omega$ in the same way as by a Dirichlet boundary condition on $\partial \Omega$. This phenomenon, a cousin of the "quantum Zeno effect," might suggest that a hard detector is mathematically impossible. Nevertheless, a mathematical description of a hard detector has recently been put forward in the form of the "absorbing boundary rule" involving an absorbing boundary condition on the detecting surface $\partial \Omega$. We show here that in a suitable (non-obvious) limit, the imaginary potential $V$ yields a non-trivial distribution of detection time and place in agreement with the absorbing boundary rule. That is, a hard detector can be obtained as a limit, but it is a different limit than Allcock considered.