Inverse Scattering Preview

Inverse scattering asks how much of an interaction can be reconstructed from scattering data. The direct scattering problem starts from a potential and computes phase shifts, amplitudes, cross sections, and bound states. The inverse problem asks whether those data determine the potential.

The distinction between measured probabilities and complex amplitudes is crucial. This introduction considers short-range quantum potentials; reconstruction theorems require a specified potential class and complete scattering data.

Reconstructing an Interaction

The direct problem is

$$V(r) \longrightarrow \{\delta_\ell(k),\text{ bound states},\text{ amplitudes}\}.$$

The inverse problem asks for the reverse:

$$\{\text{scattering data}\} \longrightarrow V(r).$$

The reverse direction is subtle because scattering experiments provide finite-precision, finite-energy, and often incomplete data. Different potentials can agree on limited data while differing outside the probed region or in off-shell behavior.

What Counts as Scattering Data

For a short-range central potential, idealized data might include:

• phase shifts $\delta_\ell(k)$ for all relevant $\ell$ and all $k>0$,
• bound-state energies,
• bound-state normalization data,
• information about long-range tails or singularities,
• channel and spin quantum numbers.

In one-dimensional or fixed-partial-wave inverse scattering, the data are often packaged in terms of reflection coefficients, bound-state poles, and norming constants.

The norming constants matter. Bound-state energies alone generally do not contain enough information to reconstruct a potential uniquely.

Uniqueness Caveats

There are uniqueness theorems under strong assumptions. But those assumptions are part of the theorem. Uniqueness can fail or become practically meaningless when:

• only finitely many phase shifts are known,
• data are available only over a finite energy range,
• bound-state normalization data are missing,
• the potential is not local or not central,
• long-range forces are present,
• inelastic channels have been integrated out,
• experimental errors are comparable to the effect being reconstructed.

Thus inverse scattering is not a license to read a unique microscopic potential directly from a cross-section plot.

Phase-Equivalent Potentials

Different potentials can produce the same phase shifts over a specified data set. Such potentials are called phase-equivalent with respect to those data.

This is not merely a technical nuisance. In effective descriptions, multiple short-distance potentials can represent the same low-energy physics. Low-energy scattering may determine parameters such as $a$ and $r_e$ while leaving short-distance structure unresolved.

A simple example already appears in the scattering-length approximation:

$$f_a(k)=\frac{1}{-1/a-ik},\qquad \sigma_a(k)=\frac{4\pi a^2}{1+k^2a^2}.$$

The cross sections for $a$ and $-a$ agree at every $k$ within this approximation, but their phases differ. Their poles are at $k=i/a$, on opposite imaginary axes. Thus even ideal measurements of this cross-section magnitude alone do not decide whether the large-$|a|$ pole is bound or virtual. Phase-sensitive information, bound-state measurements, or additional dynamical assumptions are needed.

Bound States and Poles

Bound-state data are part of inverse scattering because bound states appear as poles of the scattering amplitude. A reconstruction that uses only continuum phase shifts can miss information stored in the discrete spectrum.

For example, in one-dimensional inverse scattering, reflection data plus bound-state poles and norming constants enter reconstruction formulas. In radial three-dimensional problems, fixed-$\ell$ inverse methods similarly require bound-state information in addition to phase shifts.

Integrable-Systems Bridge

The phrase “inverse scattering” also appears in integrable systems, where scattering data for an auxiliary linear problem evolve simply in time. The Korteweg–de Vries equation is the standard example: a nonlinear wave equation is solved by mapping it to scattering data, evolving those data, and reconstructing the potential-like field.

This is related in spirit but not identical to reconstructing a quantum interaction from laboratory scattering. The shared idea is that spectral and scattering data can encode a function in a highly structured way.

Boundary with Rigorous Theory

A complete treatment of inverse scattering belongs near Sturm–Liouville theory, spectral theory, and mathematical physics. Topics include:

• Gel’fand–Levitan and Marchenko equations,
• fixed-energy versus fixed-angular-momentum inverse problems,
• uniqueness and stability theorems,
• long-range modifications,
• nonlocal and energy-dependent interactions,
• inverse problems with noise and incomplete data.

For most physics applications, the important message is practical: specify the data and assumptions before claiming a potential has been reconstructed.

Exercises

1. Why are bound-state energies alone not enough to reconstruct a potential?

Solution

Bound-state energies give only part of the spectral data. They do not determine scattering phases, reflection data, or bound-state normalization information. Many potentials can share the same finite set of bound-state energies while having different continuum scattering and different wavefunction shapes. In inverse-scattering theorems, additional data such as norming constants and phase shifts are needed.

2. Explain why low-energy scattering data cannot usually determine short-distance structure uniquely.

Solution

At low energy, the de Broglie wavelength is long compared with the potential range. Observables are controlled by a few threshold parameters such as $a$ and $r_e$. Many different short-range potentials can have the same values of those parameters, so they are indistinguishable by low-energy data even though they differ at short distance.

3. What extra assumption is hidden in the phrase “reconstruct the potential”?

Solution

One must specify the class of potentials being reconstructed: local or nonlocal, central or noncentral, short-range or long-range, single-channel or multichannel, energy-independent or energy-dependent. Without such assumptions, the inverse problem is underdetermined or not even well posed.

References

• K. Chadan and P. C. Sabatier, Inverse Problems in Quantum Scattering Theory, 2nd ed., Springer, 1989.
• V. A. Marchenko, Sturm–Liouville Operators and Applications, Birkhäuser, 1986.
• R. G. Newton, Scattering Theory of Waves and Particles, 2nd ed., Dover, 2002.