Resonances and Complex Energies

A resonance computation seeks an outgoing pole $E_\star=E_r-i\Gamma/2$, not merely an eigenvalue of a large finite box. Non-Hermitian boundary methods can expose such poles, but also generate discretized continuum eigenvalues. A credible result tracks the same candidate under independent numerical changes and relates it to the intended scattering channel.

Required background. Physical resonances and Breit–Wigner behavior explain pole interpretation. CAP and ECS supply outgoing numerical boundaries.

Energy, width, and the exponential regime

With time dependence $e^{-iEt/\hbar}$,

$$e^{-iE_\star t/\hbar} =e^{-iE_rt/\hbar}e^{-\Gamma t/(2\hbar)}, \qquad P(t)\propto e^{-\Gamma t/\hbar}.$$

Thus the probability lifetime is $\tau=\hbar/\Gamma$. Confusing amplitude decay with probability decay creates a factor-of-two error. A pole contribution dominates only over an appropriate time window; preparation transients, background continuum, and long-time threshold tails can spoil a global exponential fit.

An eigenvalue with positive imaginary part produces growth in this convention and should trigger a sign or boundary-condition check for a passive decay problem.

Track a family of eigenpairs

Compute a candidate over a parameter grid: absorber strength and onset for CAP, rotation angle and radius for ECS, plus spatial/basis resolution and domain size. Match eigenpairs by interior overlaps or physically relevant channel content, not merely by sorted real parts. Avoided crossings among complex eigenvalues can otherwise produce a spurious “stable” trajectory assembled from different states.

Inspect localization inside the interaction region together with a stable complex energy. Exterior localization or movement along the rotated continuum suggests a discretization mode. Very broad or overlapping resonances may not admit a clean single-state picture.

Real-box stabilization can identify slowly moving avoided-crossing structures, but extracting a width requires additional analysis. A Hermitian box eigenvalue has no imaginary part to interpret directly.

Residuals and nonnormal conditioning

For a computed right eigenvector $r$, the residual $\|Hr-Er\|/\|r\|$ checks the matrix eigenproblem. For a nonnormal matrix, a small residual need not imply a comparably small eigenvalue error. With left eigenvector $l^\dagger H=E l^\dagger$,

$$\delta E\approx \frac{l^\dagger(\delta H)r}{l^\dagger r}, \qquad \kappa_E=\frac{\|l\|\|r\|}{|l^\dagger r|}.$$

A large $\kappa_E$ amplifies perturbations and makes widths especially sensitive. Near an exceptional point, single-eigenvalue perturbation theory itself can fail. Report left/right normalization and condition estimates when instability appears.

If $\Gamma\ll|E_r|$, enough precision to resolve the real energy may still be insufficient for the width. Compare absolute errors in both components rather than only a relative error in $|E|$.

A pole model as an independent check

For an isolated elastic pole with negligible background variation,

$$S(E)=e^{2i\delta_{\rm bg}} \frac{E-E_r-i\Gamma/2}{E-E_r+i\Gamma/2}.$$

On the real axis, $|S|=1$, and the unwrapped resonant phase derivative at $E_r$ is $2/\Gamma$. This model tests phase unwrapping and width fitting. A varying background or a nearby threshold requires a more complete fit; the cross-section maximum need not sit exactly at the pole’s real part.

Compare the fitted width with the complex eigenvalue and, where feasible, with an intermediate-time survival-probability decay. Use consistent channel thresholds and energy origins.

Exercises

An outgoing eigenvalue is $E_\star=(0.8-0.003i)E_0$. Find its width and lifetime in natural units.

Solution

$\Gamma=0.006E_0$ and $\tau=\hbar/(0.006E_0)\approx166.7\,\hbar/E_0$. The amplitude decay time is twice this value. A width uncertainty must come from numerical and modeling variation, not from printing additional digits of the eigensolver output.

References

• 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.
• Taylor, John R., Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover Publications (2006; original edition 1972) — scattering normalization, amplitudes, and analytic structure.