Complex Absorbing Potentials

A complex absorbing potential (CAP) adds a negative imaginary term near the numerical boundary to remove outgoing amplitude. It can support wave-packet scattering and non-Hermitian resonance calculations. Its strength and shape are numerical parameters; loss caused by the absorber is not automatically a physical decay width.

Required background. Absorbing boundaries explains wave-packet boundary control, and resonances defines physical decay poles.

The sign of absorption

Write

$$H_\eta=H-i\eta W,\qquad \eta>0,\quad W\geq0,$$

with $W$ supported outside the interaction region. For real-time evolution,

$$\frac{d}{dt}\langle\psi|\psi\rangle =-\frac{2\eta}{\hbar}\langle\psi|W|\psi\rangle.$$

The expectation here uses the current unnormalized state. Reversing the sign turns damping into amplification. Renormalizing at every time step destroys the physical interpretation of absorbed probability.

Record inner-region probability, outgoing flux, absorbed probability, and any remaining exterior probability. Their balance distinguishes intended absorption from numerical probability loss or reflected waves.

Stronger absorption can reflect more

For a sharp imaginary step $-i\eta W_0$ and positive incident energy $E$, the absorber wave number is

$$q=\sqrt{\frac{2\mu}{\hbar^2}(E+i\eta W_0)}, \qquad \operatorname{Im}q>0.$$

Continuity gives $r=(k-q)/(k+q)$. As $\eta W_0$ becomes extremely large, $|r|$ approaches one: a strong abrupt absorber behaves like a reflecting wall. A smooth turn-on reduces the mismatch, while sufficient layer width allows attenuation before the outer boundary.

Calibrate the layer with free packets spanning the momentum range used in the physical run. Very low momentum typically requires particular care because its wavelength is long relative to a fixed ramp. A single well-absorbed central momentum does not test the packet tails.

A wave-packet convergence protocol

Hold the physical potential and initial packet fixed. Increase the inner-region size so the absorber begins outside the interaction and observable-support region. Then vary absorber onset, width, strength, and smoothness. Refine the grid and time step independently.

Measure reflected probability after a packet crosses the layer. Confirm that the wave packet has not yet returned from the computational boundary when evaluating scattering observables. A useful flux budget includes both transmitted and reflected detectors placed before the absorber; integrate until the relevant packet has passed them.

CAP-induced loss is appropriate for outgoing boundary control. It does not model an arbitrary physical environment or supply a complete quantum channel without an additional model.

Resonance trajectories in a finite basis

Diagonalizing $H_\eta$ gives complex eigenvalues that move with $\eta$. A physical resonance candidate often has a comparatively stationary region as strength and basis parameters vary, while discretized continuum modes move strongly. Track states using overlaps or interior localization, since sorting by real energy alone can exchange identities.

Compare $\operatorname{Re}E$ and $\Gamma=-2\operatorname{Im}E$ under basis refinement, layer changes, and another outgoing-boundary method. Small $|\eta\,dE/d\eta|$ is a stationarity diagnostic, not a proof of an exact pole.

At fixed finite Hermitian basis, sending $\eta\to0$ returns real eigenvalues. Thus the desired outgoing limit cannot be inferred by a naive zero-strength extrapolation at fixed basis. The basis and absorber limits must be coordinated.

Exercises

For constant $W=W_0$ everywhere and an initially normalized eigenstate of $H$, find the norm and squared norm and explain why their decay does not measure a resonance.

Solution

The amplitude and norm acquire the same factor, $\|\psi(t)\|=e^{-\eta W_0t/\hbar}$. The squared norm, which gives the remaining probability, is $\|\psi(t)\|^2=e^{-2\eta W_0t/\hbar}$. This decay is imposed uniformly by the chosen absorber and occurs even for a truly bound state. A physical resonance requires outgoing-wave dynamics and stability under changes of the artificial boundary treatment.

References

• Manolopoulos, David E., “Derivation and reflection properties of a transmission-free absorbing potential,” The Journal of Chemical Physics 117, 9552–9559 (2002), doi:10.1063/1.1517042 — absorber design and reflection analysis.
• Moiseyev, Nimrod, “Quantum theory of resonances: calculating energies, widths and cross-sections by complex scaling,” Physics Reports 302, 212–293 (1998), doi:10.1016/S0370-1573(98)00002-7 — non-Hermitian resonance calculations.