Poles, Resonances, and Scattering Channels

A collision can reveal a bound state, a long-lived resonance, or the opening of a new internal channel. The connection becomes clearer when scattering amplitudes are continued away from real collision energies. Their poles describe different kinds of states, while thresholds determine which outgoing motions are physically possible.

Helpful background. The S-matrix organizes incoming and outgoing channels; partial waves and thresholds introduces phase shifts and low-energy parameters.

Recognize three kinds of pole

Start with bound states and scattering poles. In a short-range, single-channel problem, an outgoing radial wave has the large-distance dependence $e^{ikr}$. Continuing $k$ to complex values distinguishes solutions with different spatial behavior.

Pole type	Typical momentum location	Interpretation
Bound state	$k=i\kappa$, $\kappa>0$	The continued outgoing wave decays at infinity and can be normalizable.
Virtual state	$k=-i\kappa$, $\kappa>0$	The continued wave grows at infinity; this is not a normalizable bound state.
Decaying resonance	$k=k_R-ik_I$, with $k_R,k_I>0$	An outgoing resonance solution describes a decaying pole contribution in time.

This table is a single-channel guide. In a multichannel problem, every threshold introduces a channel momentum and a branch choice, so the energy sheet must be specified before interpreting a pole.

A resonance is more than a peak

Resonances relates poles to phase motion, time delay, and energy-dependent cross sections. For a simple isolated decaying pole, write

$$E_p=E_R-i\Gamma/2,\qquad \Gamma>0.$$

Here $\Gamma$ has energy units. Its pole contribution evolves as $e^{-iE_Rt/\hbar}e^{-\Gamma t/(2\hbar)}$, so the probability lifetime is $\tau=\hbar/\Gamma$. A wave packet also contains background contributions, and exact decay need not be exponential for all times.

The Breit–Wigner form is useful when a sufficiently isolated pole and slowly varying background dominate the relevant energy interval. Threshold factors, background interference, or overlapping resonances can shift the cross-section maximum and spoil a symmetric line shape. A phase crossing, a maximum, a fitted width, and a pole parameter are therefore quantities to compare, not automatically interchangeable definitions.

The square-well resonance example makes these differences concrete by comparing real-energy scattering with an outgoing-wave pole calculation.

Open and closed channels play different roles

Multichannel scattering introduces several asymptotic internal states. If channel $\beta$ has threshold energy $\varepsilon_\beta$, its kinetic energy is $E-\varepsilon_\beta$. An open channel can carry outgoing flux. A closed channel has an evanescent asymptotic solution, but coupling through it can still shift levels or produce a resonance in an open channel.

The full open-channel S-matrix conserves probability under the usual closed-system assumptions and consistent flux normalization. Its elastic subblock need not be unitary because probability can emerge in other channels.

Inelastic scattering follows the energy transfer and observable channel rates. In unit incident-wave conventions, the conversion from amplitude to differential cross section includes the outgoing-to-incoming speed ratio. A threshold can close a channel even when its coupling matrix element remains nonzero.

What can be inferred from measured scattering?

Inverse scattering asks what properties of an interaction can be recovered from scattering information. Differential cross sections determine amplitude magnitudes and generally do not provide every phase. Uniqueness depends on what is measured and on assumptions about the potential, energy range, and available bound-state data.

The bridge to QFT scattering preserves channels, thresholds, and pole analysis while adding relativistic kinematics and particle creation. Continue there after the nonrelativistic distinctions above are clear.

Exercises

Width versus decay rate. A resonance has energy width $\Gamma=0.1\hbar\omega$. What are its probability decay rate and lifetime in the pole approximation?

Solution

The rate is $\gamma=\Gamma/\hbar=0.1\omega$, and $\tau=1/\gamma=10/\omega$. The amplitude decays at half this rate. Confusing an energy width with an inverse time omits a factor of $\hbar$.

A closed channel. A system has thresholds $0$ and $\varepsilon>0$, and collision energy $0<E<\varepsilon$. Does the second channel have no effect?

Solution

It carries no asymptotic outgoing flux, but virtual coupling to it can modify the first-channel scattering. If a state associated with the closed channel approaches the collision energy and couples to the open channel, it can produce a resonance. Removing a closed channel without accounting for its virtual effect may therefore miss important physics.

References

• Chin, C., R. Grimm, P. Julienne, and E. Tiesinga. “Feshbach resonances in ultracold gases.” Reviews of Modern Physics 82, 1225–1286 (2010). doi:10.1103/RevModPhys.82.1225. Open and closed channels and tunable threshold resonances.
• Newton, R. G. Scattering Theory of Waves and Particles. 2nd ed., Dover (2002; reprint of the 1982 edition). Analytic continuation and inverse scattering.
• Taylor, J. R. Scattering Theory: The Quantum Theory of Nonrelativistic Collisions. Dover (2006; reprint of the 1972 edition). Poles, resonances, and multichannel scattering.