Scattering Overview

Mathematical scattering asks whether interacting motion can be described, at large positive or negative times, by a simpler comparison dynamics. The central issue is a limit of normalizable wave packets. Cross sections and stationary amplitudes are consequences of that construction together with additional regularity and normalization choices. This page organizes the distinct claims that must be proved before those consequences can be used.

Helpful background. Stone’s Theorem constructs unitary dynamics. Spectral Projections separates invariant spectral sectors. Scattering States distinguishes packets from generalized kets.

Scattering compares two complete motions

Let $H_0$ and $H$ be self-adjoint operators on a Hilbert space, with evolutions $U_0(t)=e^{-itH_0/\hbar}$ and $U(t)=e^{-itH/\hbar}$. An incoming comparison vector $\phi_{\rm in}$ represents an interacting vector $\Psi$ if

$$\|U(t)\Psi-U_0(t)\phi_{\rm in}\|\longrightarrow0 \qquad(t\to-\infty).$$

The outgoing condition uses $t\to+\infty$ and generally a different vector $\phi_{\rm out}$. The interaction need not stop at a finite time. The assertion concerns an asymptotic norm difference, which controls every bounded measurement uniformly in operator norm.

Indeed, for normalized $\psi_t,\phi_t$ with norm difference $\epsilon_t$, and bounded $A$,

$$|\langle\psi_t,A\psi_t\rangle-\langle\phi_t,A\phi_t\rangle| \leq2\|A\|\epsilon_t.$$

There is no corresponding conclusion for arbitrary unbounded observables without additional moment control. A tiny norm tail may carry a large energy or position moment.

Existence and exhaustiveness answer different questions

Wave Operators makes the comparison precise on the absolutely continuous subspace of $H_0$. Existence says that each admissible free packet determines an interacting packet. It does not say that every interacting continuous-spectrum packet arises this way.

Asymptotic Completeness asks whether the wave-operator range equals $\mathcal H_{\rm ac}(H)$. This convention concerns the absolutely continuous part. Some references reserve “asymptotic completeness” for exhaustion of the entire continuous subspace. Those conventions agree when a separate theorem excludes singular continuous spectrum.

Three checks must therefore remain distinct:

• the comparison limits exist on the intended initial space;
• their ranges exhaust the stated interacting space;
• any spectral sector omitted from that space is absent or is treated separately.

The pure point subspace includes embedded eigenvectors as well as isolated bound levels. The location of an eigenvalue inside a continuum does not turn its normalizable eigenvector into an absolutely continuous scattering packet.

Local escape is weaker than asymptotic freedom

A useful spectral diagnostic is the RAGE theorem. If $C$ is compact and $\Psi$ belongs to the continuous spectral subspace of a self-adjoint $H$, then

$$\frac1T\int_0^T\|CU(t)\Psi\|^2\,dt\longrightarrow0.$$

This is an averaged statement. It supplies neither a decay rate nor a single free packet approximating the interacting orbit. For the absolutely continuous sector, compactly tested decay also holds pointwise in time; singular continuous spectral measures do not support that general inference.

Spatial localization itself is not compact on $L^2(\mathbb R^d)$. To apply RAGE to a particle model, one typically combines localization with an energy cutoff or a resolvent, proves that this combination is compact, and controls the remaining spectral tail. Omitting this compactness step changes the theorem.

Long-range forces give another reason escape is insufficient. Coulomb packets may escape to infinity while accumulating a logarithmic phase relative to free motion. Their comparison needs Long-Range Potentials, even though their spatial separation becomes large.

Two complementary methods

The time-dependent approach estimates trajectories. Cook’s Method integrates the interaction acting on freely evolving packets. Stronger propagation arguments establish completeness by controlling arbitrary interacting packets.

The stationary approach studies energy boundary values of resolvents. Stationary Scattering Theory explains why the limit requires weighted spaces. Resolvent Methods connects those estimates to spectral densities and perturbation factorizations.

Neither approach licenses substitution of an exact-energy plane wave into an $L^2$ theorem. One first establishes packet operators or a justified spectral transform. Distributional kernels then express those operators in a generalized basis.

Resonances are a further analytic question

A self-adjoint Hamiltonian has real spectrum. A complex resonance belongs to a continued resolvent or an associated nonself-adjoint deformation, not to its ordinary Hilbert-space eigenvalue equation. Continuation needs hypotheses beyond self-adjointness, and its parameter surface depends on dimension and channel thresholds.

Resonances gives a controlled compact-support model. The physical interpretation through phase shifts and line shapes belongs to Scattering Resonances. Neither a peak nor an apparently exponential time trace alone establishes a pole theorem.

Exercises

1. Prove the bounded-observable estimate above.

Solution

Insert and subtract $\langle\phi_t,A\psi_t\rangle$. Cauchy–Schwarz bounds the resulting terms by $\|A\|\|\psi_t-\phi_t\|\|\psi_t\|$ and $\|A\|\|\phi_t\|\|\psi_t-\phi_t\|$. Normalization gives the factor two. Taking $A$ to be an event projection gives a probability bound.

2. Does absence of eigenvalues establish a complete free scattering description?

Solution

No. It only eliminates the pure point sector. Singular continuous spectrum could remain, the chosen wave operators might not exist, or their ranges might not exhaust the absolutely continuous sector. Even a purely absolutely continuous Hamiltonian requires a comparison theorem for the specified $H_0$.

References

• G. Teschl, Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators, 2nd ed., Graduate Studies in Mathematics 157, American Mathematical Society, 2014, Sections 5.2 and 12.1–12.3, author’s text — spectral escape and dynamical comparison.
• D. Yafaev, “Lectures on scattering theory,” 2004, arXiv:math/0403213, Lectures 1–3 — time-dependent, stationary, and long-range constructions.