Absorbing Boundaries

An absorbing boundary approximates escape from a finite computational domain. The target is accurate evolution in an interior region over a stated observation time and momentum range. A boundary can remove almost all probability while still reflecting enough amplitude to corrupt a small signal; absorption efficiency and reflection error therefore require separate measurements.

Required background. Boundary conditions distinguishes finite-box domains from outgoing problems. Wave-packet propagation explains resolution and flux checks. Helpful background. Complex absorbing potentials derives the norm-loss identity, reflection from a sharp imaginary step, and resonance diagnostics.

A free exterior produces a memory boundary condition

Consider the one-dimensional Schrödinger equation beyond a right boundary $x=b$, with constant mass, zero exterior potential, and initially zero wavefunction in that exterior. Define $\alpha=\hbar^2/(2m)$. A Laplace transform in time, with $\operatorname{Re}s>0$, gives

$$\partial_x^2\widehat\psi =-\frac{i\hbar s}{\alpha}\widehat\psi.$$

The solution decaying toward $+\infty$ is proportional to $e^{-\lambda(s)(x-b)}$, where

$$\lambda(s)=e^{-i\pi/4}\sqrt{\frac{2m}{\hbar}}\sqrt{s}, \qquad \operatorname{Re}\lambda>0.$$

Consequently $\partial_x\widehat\psi(b,s)=-\lambda(s)\widehat\psi(b,s)$. In time this becomes

$$\partial_x\psi(b,t) =-e^{-i\pi/4}\sqrt{\frac{2m}{\hbar}}\, D_t^{1/2}\psi(b,t),$$

where, for the compatible zero-initial boundary trace,

$$D_t^{1/2}g(t)=\frac1{\sqrt\pi}\frac{d}{dt} \int_0^t\frac{g(\tau)}{\sqrt{t-\tau}}\,d\tau.$$

The boundary remembers earlier outgoing amplitude. At a left boundary the derivative sign reverses; writing the condition using the outward normal makes the two cases uniform. A nonzero initial exterior or a nontrivial exterior potential requires a modified condition, not the formula above with unchanged data. Antoine and collaborators review these transparent and approximate boundary constructions.

Continuum transparency is not discrete transparency

A discretized interior equation has its own dispersion relation. Combining a continuum radiation condition with that discretization can introduce reflection even when both pieces are individually consistent. A discrete transparent boundary is derived from the same spatial and temporal scheme as the interior update; its convolution weights and initialization depend on that scheme.

Local outgoing approximations avoid the full time history by matching a restricted frequency range. A first-order condition tuned to one wave number cannot be exactly transparent for every Schrödinger frequency. Test broad packets and low-momentum tails instead of relying solely on a central-frequency example.

Masks and absorbing layers

A practical alternative appends an exterior layer, propagates the state, and multiplies grid values by a real mask $0<M(x)\leq1$. Over a time step $\Delta t$, this damping substep is the exact exponential of a local imaginary potential $-iW$ with

$$W(x)=-\frac{\hbar}{\Delta t}\log M(x)\geq0.$$

If the time step changes while the mask is held fixed, the effective physical absorption strength changes. To compare temporal refinements at fixed $W$, use $M_{\Delta t}=e^{-W\Delta t/\hbar}$. A symmetric damping–Hamiltonian–damping composition can reduce splitting error when its regularity assumptions hold; a one-sided mask application is generally only first-order as a combined noncommuting splitting.

Turn the layer on outside the interaction region and all detectors used for the observable. Its thickness and smoothness control reflection as well as attenuation. Increasing strength without widening or smoothing the layer can make reflection worse, as the CAP step calculation demonstrates. A perfectly matched layer instead modifies the differential operator through a complex coordinate construction; it is not simply another name for arbitrary damping.

Probability accounting and calibration

For a local sink $W\geq0$, the integrated absorbed probability is

$$P_{\mathrm{abs}}(t)=\frac2\hbar\int_0^t \langle\psi(\tau)|W|\psi(\tau)\rangle\,d\tau.$$

The expectation uses the unnormalized state. With a closed outer boundary and exact dynamics, $\|\psi(t)\|^2+P_{\mathrm{abs}}(t)=\|\psi(0)\|^2$. A mask-only damping substep permits the direct discrete measurement $\sum_j w_j(1-M_j^2)|\psi_j|^2$ without approximating that substep’s lost norm by a time quadrature.

For scattering, place flux detectors before the layers and integrate until the outgoing packets have crossed them. Compare detector flux, retained norm, and absorbed norm. Absorbed norm alone cannot assign a physical channel unless geometry and detector definitions justify that assignment. Do not renormalize after absorption when extracting absolute probabilities.

Calibrate with free packets spanning the actual momentum distribution. Vary domain extent, absorber onset, width, and strength independently of grid and time-step refinement. Measure any returning packet against a larger-domain reference over the same time window. The chosen tolerance should be smaller than the observable being resolved: a reflection probability of $10^{-4}$ is unacceptable when measuring transmission of order $10^{-6}$.

Exercises

A mask $M=0.9$ is applied once per step to an initially normalized state in a region with no transport. What are the norm and remaining probability after ten steps? What mask gives the same damping rate when the step is halved?

Solution

The norm scales like the amplitude: $\|\psi\|=0.9^{10}\approx0.348678$. The remaining probability is the squared norm, $\|\psi\|^2=0.9^{20}\approx0.121577$. The half-step mask is $\sqrt{0.9}$. Twenty half steps then produce the same amplitude and probability. Reusing $0.9$ would double the decay rate per unit physical time.

Why is zero measured reflected flux insufficient if almost all of a packet remains inside the absorber at the end of the run?

Solution

The measurement may end before delayed reflection or return from the outer boundary reaches the detector. Extend the observation window or enlarge the domain and account explicitly for retained exterior probability. Transparency is a statement about the interior solution over a specified interval, not only an early-time flux sample.

References

• Antoine, Xavier, Anton Arnold, Christophe Besse, Matthias Ehrhardt, and Achim Schädle, “A Review of Transparent and Artificial Boundary Conditions Techniques for Linear and Nonlinear Schrödinger Equations,” Communications in Computational Physics 4, 729–796 (2008). Publisher article.
• Manolopoulos, David E., “Derivation and Reflection Properties of a Transmission-Free Absorbing Potential,” Journal of Chemical Physics 117, 9552–9559 (2002). DOI.