Bound States and Scattering Poles

Scattering amplitudes remember the spectrum. Bound states, virtual states, and resonances appear as poles of analytically continued scattering amplitudes or $S$-matrix elements. The physical distinction among them is encoded in where the pole sits in the complex energy or complex momentum plane. The residue of an isolated bound-state pole in the coordinate resolvent is derived in Energy Green Function. For the underlying analytic-function language, see Complex Analysis Essentials; for sheets and threshold cuts, see Branch Cuts.

For one-channel nonrelativistic $s$-wave scattering,

$$f_0(k) = \frac{1}{k\cot\delta_0(k)-ik}.$$

Poles occur when the denominator vanishes after analytic continuation away from real positive $k$.

Energy and Momentum Planes

For relative motion with reduced mass $\mu$,

$$E = \frac{\hbar^2k^2}{2\mu}.$$

Positive real $k$ describes physical elastic scattering above threshold. Negative energy corresponds to imaginary $k$:

$$k=i\kappa \quad \Rightarrow \quad E = - \frac{\hbar^2\kappa^2}{2\mu}.$$

Analytic continuation introduces different sheets of the energy plane. The momentum plane is often the cleaner language near a single threshold. Choose the physical energy sheet by $\operatorname{Im}k>0$, with $k>0$ on the upper rim of the positive-energy cut. The lower half of the $k$ plane describes the second sheet. A bound pole at $+i\kappa$ and a virtual pole at $-i\kappa$ have the same negative value of $E$, but lie on different sheets.

Bound-State Poles

A normalizable bound state has

$$k=i\kappa, \qquad \kappa\gt0.$$

The radial wave outside the range of the potential decays as

$$e^{-\kappa r}.$$

Thus the pole lies on the positive imaginary $k$ axis. Its binding energy is

$$E_b = \frac{\hbar^2\kappa^2}{2\mu},$$

so the energy eigenvalue is $-E_b$ relative to threshold.

For a short-range interaction of range $R$, with $a\gg R$ and $|r_e|\ll a$, a shallow $s$-wave bound state satisfies

$$\kappa\approx\frac1a, \qquad E_b \approx \frac{\hbar^2}{2\mu a^2}.$$

This is the pole interpretation of the large-$a$ universal formula. Positive $a$ by itself is insufficient: a hard sphere has $a=R>0$ and no bound state. The shallow-pole inference requires the separation of scales just stated.

Virtual States

A virtual state is a pole near threshold on the negative imaginary $k$ axis:

$$k=-i\kappa, \qquad \kappa\gt0.$$

Its analytically continued outgoing wave is $e^{ikr}=e^{+\kappa r}$. It is not a normalizable bound state; the pole lies on the second energy sheet. Nevertheless, it can strongly affect low-energy scattering.

Large negative scattering length is often associated with a virtual state near threshold. This is why a negative $a$ can still signal strong low-energy attraction.

Resonance Poles

A resonance corresponds to a pole away from the imaginary axis on an unphysical sheet. In the energy plane it is commonly written

$$E_{\mathrm{pole}} = E_R - \frac{i}{2}\Gamma,$$

where $E_R$ is the resonance energy and $\Gamma$ is the width. The associated lifetime scale is

$$\tau\sim\frac{\hbar}{\Gamma}.$$

For real scattering energies, the same pole can appear as a rapid phase variation, a time delay, and a peak or distortion in a cross section. Resonances treats the physical classification, while Breit–Wigner Form derives the isolated-pole line shape and its limits. Resonance from a Square Well verifies this correspondence by comparing a real-axis fit with the exact outgoing-wave pole.

Effective-Range Pole Equation

Low-Energy S-Wave Scattering compares this truncated pole equation with the exact analytic continuation of a tuned square well and identifies the second, range-scale root as an artifact of extrapolating the expansion too far.

Using

$$k\cot\delta_0(k) = - \frac1a + \frac12r_ek^2 + \cdots,$$

the pole condition

$$k\cot\delta_0(k)-ik=0$$

at $k=i\kappa$ gives

$$- \frac1a - \frac12r_e\kappa^2 + \kappa + \cdots =0.$$

This equation shows how scattering length and effective range encode the location of a near-threshold bound-state pole.

S-Matrix View

For one-channel elastic scattering,

$$S_\ell(k)=e^{2i\delta_\ell(k)}$$

on the physical real axis. Analytically continuing $S_\ell(k)$ reveals poles and zeros. Bound-state poles on the physical sheet correspond to discrete normalizable states. Resonance poles usually lie on unphysical sheets reached by continuing through a branch cut.

The phrase “pole of the S-matrix” is therefore shorthand for a statement about analytic continuation, not merely a divergence at a real laboratory energy. Delta Potential Scattering supplies an exact one-dimensional example in which attraction moves the pole to $k=i\kappa$, repulsion moves it to $k=-i\kappa$, and no off-axis resonance pole appears.

QFT Bridge

The same analytic idea survives in relativistic scattering. Stable particles and bound states appear as poles of amplitudes on physical sheets. Unstable particles and resonances appear as poles at complex invariant mass on appropriate unphysical sheets.

The details change: relativistic normalization, many-particle thresholds, spin, crossing, and field-theoretic branch cuts matter. But the central lesson is shared: spectrum and scattering are different boundary values of the same analytic structure.

Exercises

1. Classify the pole $k=i\kappa$ with $\kappa\gt0$.

Solution

The energy is

$$E = - \frac{\hbar^2\kappa^2}{2\mu}.$$

The outside radial behavior decays as $e^{-\kappa r}$. This is a normalizable bound state below threshold.

2. Use the scattering-length approximation to find the pole of $f_0(k)$.

Solution

The scattering-length approximation is

$$f_0(k) = \frac{1}{-1/a-ik}.$$

The pole satisfies

$$- \frac1a - ik =0,$$

so

$$k=\frac{i}{a}.$$

Within this approximation, $a>0$ gives a bound pole and $a<0$ a virtual pole. These predictions describe the original interaction only if $|a|$ is large compared with its range and effective range, so the predicted $|k|=1/|a|$ lies inside the expansion’s domain.

3. Why is analytic continuation necessary to define a resonance pole?

Solution

For real energies in an elastic channel, unitarity keeps the physical $S$-matrix finite. A resonance affects the real axis through phase motion and line shapes, but its pole is at complex energy, typically $E_R-i\Gamma/2$, on an unphysical sheet. Reaching that point requires analytic continuation through the branch structure of the amplitude.

References

• H. A. Bethe, “Theory of the effective range in nuclear scattering,” Physical Review 76, 38–50 (1949), doi:10.1103/PhysRev.76.38.
• G. Breit and E. Wigner, “Capture of slow neutrons,” Physical Review 49, 519–531 (1936), doi:10.1103/PhysRev.49.519.
• C. J. Joachain, Quantum Collision Theory, 3rd ed., North-Holland, 1983.
• R. G. Newton, Scattering Theory of Waves and Particles, 2nd ed., Dover, 2002.
• J. R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover, 2006.