Absorbing boundaries and reflection artifacts

A common model adds minus i times a nonnegative Hermitian operator W to an otherwise Hermitian Hamiltonian. For a spatial absorbing potential, W is a nonnegative real function supported near a boundary. Differentiate the ordinary squared state norm using the Schrödinger equation and its adjoint. The Hermitian terms cancel and the absorbing contribution is nonpositive. This identity explains intentional loss without calling the finite model unitary.

The lost norm is a numerical bookkeeping quantity unless a specific physical measurement or reservoir model justifies another interpretation. A complex absorbing potential is not, by itself, a trace-preserving Lindblad equation for a reduced density matrix. If the remaining state is renormalized, it becomes a conditional description of surviving represented amplitude; the removed weight must still be recorded when computing unconditional reflection or transmission probabilities.

A very abrupt absorbing layer can itself reflect a wave, while a weak or short layer can leave substantial amplitude to reach the outer boundary. Smoothness, strength, thickness and the outgoing energy band therefore need joint testing. Increasing strength monotonically does not necessarily improve the reflection error. Transparent boundary conditions, complex scaling and masks use different constructions with their own assumptions; they should not be treated as identical algorithms merely because they suppress returning signals.

A multiplicative mask removes amplitude at discrete times. Its probability effect involves the square of its magnitude, and its relationship to a continuous W depends on timestep. Changing the timestep while keeping an arbitrary per-step mask fixed can therefore change the physical loss rate of the numerical model. A sensible convergence study either holds the intended generator fixed or explicitly studies the resulting change in absorber prescription.

The time at which a reflected packet center returns to an interaction region can be estimated from travel distance and a stipulated group speed. This gives a useful diagnostic when moving the outer boundary: a corresponding shift of a spurious feature can reveal its boundary origin. A nonrelativistic Schrödinger packet has dispersive tails, however, so a center-travel estimate is not a strict causal guarantee that every earlier sample is uncontaminated.

Compare calculations with changed box size and absorber profile while holding the desired incident state and interaction fixed. Monitor the region of interest, outgoing flux and accumulated loss separately. A tiny final norm does not prove negligible reflection; the lost amplitude could have included desired components or masked an error after it occurred. Benchmarks with known scattering solutions and parameter-independent observables are more informative than judging an absorber from an attractive fading animation.