Mathematical Scattering Open Problems

Mathematical scattering research studies the relation between interacting dynamics and asymptotically simpler channels. Many short-range existence and completeness theorems are established. A new project must specify the feature that lies outside the theorem being used: thresholds, long-range tails, channel structure, singularity, or quantitative error. Reviewed 11 October 2026, this page uses Yafaev’s lectures and selected threshold results through 2015 as its explicit theoretical baseline, rather than a comprehensive current list of unsolved scattering problems.

Helpful background. Asymptotic completeness and wave operators distinguish existence from exhaustion of scattering states.

State the comparison dynamics

In a standard short-range setting,

$$W_\pm= \operatorname*{s-lim}_{t\to\pm\infty} e^{itH}e^{-itH_0}P_{\mathrm{ac}}(H_0).$$

A long-range interaction may require modified comparison dynamics rather than the free evolution shown here. Failure of this unmodified limit is not proof that physical scattering is impossible.

Completeness is a range statement. Unitarity of a scattering matrix on the chosen channel space alone does not prove that every relevant physical state is included. Many-body channels, bound clusters, and thresholds must be identified explicitly.

For a concrete sufficient existence test, suppose $H_0$ is self-adjoint, $V$ is bounded and self-adjoint, and $H=H_0+V$ on $D(H_0)$. Assume

$$\int_0^\infty\|V e^{-itH_0}\psi\|\,dt<\infty$$

on a dense set in the absolutely continuous subspace of $H_0$. Differentiating $e^{itH}e^{-itH_0}\psi$ and integrating makes this family Cauchy, giving the positive-time wave operator; the negative-time argument is analogous. This Cook argument does not by itself establish completeness. Failure of its integrability test is also not a nonexistence theorem. Along a ballistic ray, a Coulomb tail produces a phase accumulating like $\int^t ds/s$, explaining why a modified reference phase can be required. That ray argument motivates a modifier; proving its validity requires propagation estimates.

A threshold where the order of limits is visible

Choose units with $\hbar=1$ and $2m=1$, and the one-dimensional square well $V(x)=-V_0$ on $|x|<a$, zero outside, with $V_0>0$. For incoming momentum $k>0$, let $q=\sqrt{k^2+V_0}$. Matching the wavefunction and its derivative at both edges gives the transmission probability

$$T(k)=\left[1+ \frac{V_0^2\sin^2(2aq)}{4k^2q^2}\right]^{-1}.$$

Generically $T(k)\to0$ as $k\to0$. If $2a\sqrt{V_0}=r\pi$ for a positive integer $r$, then $\sin(2aq)=O(k^2)$ and $T(k)\to1$. At this tuned depth there is a bounded zero-energy solution that is not square integrable, called a threshold resonance in this setting. It is not an additional normalizable bound state.

Near that depth, define $\delta=2a\sqrt{V_0}-r\pi$. The leading small-$k$ denominator contains $V_0\delta^2/(4k^2)$. Thus taking $k\to0$ at fixed nonzero detuning and taking detuning to zero first give different limits. A fit made only at one depth can miss the crossover scale $k\sim\sqrt{V_0}|\delta|$.

Established context. Egorova, Holzleitner, and Teschl treat low-energy scattering and regularity under integrable weighted moments of the potential, including resonant cases. Their general theory explains why the threshold hypotheses deserve separate attention. The square-well calculation is a solvable illustration, not a claim that threshold scattering in one dimension remains generally unsolved.

Directions with a clear mathematical obstacle

Threshold behavior — active quantitative direction. Zero-energy resonances can change low-energy expansions and decay estimates. Determine how a specified threshold singularity affects an observable or approximation, after identifying what the applicable weighted-potential theorem already proves.

Long-range modifiers. Quantify the error introduced when a Coulomb-like tail is cut off or when a free reference is used in a finite-time simulation.

Channel completeness. For a restricted model, identify all asymptotic channels and prove that no component remains outside them. Do not describe established short-range many-body completeness as a universally open problem.

Certified resonances. Nonselfadjoint deformations and absorbing boundaries can expose resonances but also introduce spurious eigenvalues. A stable plot under one parameter sweep is not a proof of the resonance’s location.

Yafaev supplies the conceptual framework. Spectral-computation results clarify why finite data and truncations require their own guarantees.

An entry project: finite-time scattering near a threshold

Baseline. Use the square well above with $a=1$ and tune the depth through $V_0=\pi^2/4$. Reproduce the stationary transmission formula by matching, then propagate incoming wavepackets first away from and then close to zero energy. This controls the model, threshold, and reference answer from the beginning.

Resources. A laptop, Fourier analysis, stationary scattering, and numerical convergence methods. Use two independent descriptions, such as stationary matching and time propagation.

Deliverable. Quantify errors due to finite propagation time, box size, packet bandwidth, and boundary absorption. Then tune a parameter toward a threshold resonance and identify which error dominates.

Compare the separated transmitted packet probability with $\int_0^\infty |\phi(k)|^2T(k)\,dk$, where $\phi$ is the normalized positive-momentum incoming amplitude, rather than automatically with $T(k_0)$. For a packet supported in $[k_0-\Delta k,k_0+\Delta k]\subset(0,\infty)$, a direct mean-value estimate bounds that replacement error by $\Delta k\sup|T'(k)|$ on the interval. This bound itself can worsen near the crossover; narrowing the packet increases its spatial extent and hence the box and propagation requirements. The group velocity is $2k$ in the chosen units, so a fixed propagation time is especially misleading near zero energy.

Failure criterion. If the packet has not separated into asymptotic regions, its spatial probability split is not yet a scattering coefficient. If loss at an absorber is mistaken for transmission, the comparison is invalid.

Research extension. Prove an explicit convergence rate for the chosen model or construct an asymptotic correction that remains uniform near the threshold. A controlled restricted theorem is a meaningful contribution even if the full scattering theory is known.

Finish the first stage with an error budget that bounds the bandwidth contribution analytically and demonstrates independent time, spatial-domain, and grid convergence. Stop claiming a threshold effect when these errors exceed the difference between resonant and detuned predictions. A research continuation could seek a uniform finite-time estimate for a specified potential family and incoming momentum window. It must compare against existing dispersive and low-energy estimates, and say whether it improves regularity assumptions, constants, or parameter uniformity. Replotting the square-well formula is not original research.

Choosing a feasible theorem

Start with one geometry and one asymptotic obstruction. Before calling a problem open, check the exact hypotheses of the strongest applicable theorem and the specialized literature for that model. The proposed numerical baseline helps locate a useful regime; it does not establish novelty.

A result changes the assessment when it removes a hypothesis, identifies a missing channel, or provides a usable uniform error estimate. For numerical work, a certified resonance or scattering quantity is stronger than a larger computation with unquantified boundaries.

References

• J. Ben-Artzi, M. J. Colbrook, A. C. Hansen, O. Nevanlinna, and M. Seidel, “Computing Spectra — On the Solvability Complexity Index Hierarchy and Towers of Algorithms,” arXiv:1508.03280 (2015; see version history). Spectral certification framework.
• I. Egorova, M. Holzleitner, and G. Teschl, “Zero Energy Scattering for One-Dimensional Schrödinger Operators and Applications to Dispersive Estimates,” Proceedings of the American Mathematical Society, Series B 2, 51–59 (2015). Threshold analysis.
• D. R. Yafaev, “Lectures on scattering theory,” arXiv:math/0403213 (2004), based on 2001 lectures. Author’s notes.

Entry points

Read the wave-operator hypotheses and identify the relevant channel space. Reproduce a solvable scattering model, including its threshold limit, before proposing an extension.