Mathematical Scattering

Scattering theory compares an interacting evolution with a simpler asymptotic motion. This chapter develops the mathematical conditions behind that comparison: existence of wave operators, exhaustion of the scattering sector, resolvent boundary values, and analytic continuation. It separates exact theorems from sufficient estimates and explicitly marked proof outlines.

The Scattering Theory volume develops preparation and detection, amplitudes, flux normalization, cross sections, partial waves, and line shapes. The present chapter supplies rigorous hypotheses and proofs for the operator and spectral claims used there. Scattering States explains generalized scattering kets and their test-space interpretation.

Enter this chapter

Start with Scattering Overview to distinguish existence, completeness, spectral escape, and resonance continuation. It explains why local escape alone does not supply a free asymptote.

The main background is self-adjointness, spectral measures, and strongly continuous unitary evolution. Stone’s Theorem supplies the dynamics; Spectral Projections and Resolvent Operators supply the spectral language. The chapter introduces additional weighted spaces when they become necessary.

Construct and identify the scattering maps

1. Free and Interacting Dynamics studies comparison operators, accumulated phases, and the role of reference channels. A constant energy offset shows why small bounded perturbations alone do not guarantee convergence.
2. Wave Operators proves isometry, reducing-range properties, spectral intertwining, and the exact transport of generator domains.
3. S-Matrix proves a range-defect criterion for unitarity and explains the almost-everywhere meaning of energy-shell matrices.
4. Asymptotic Completeness proves the reverse-wave-operator criterion and gives a comparison whose scattering operator is unitary although an entire interacting continuum is missing.

Incoming and outgoing labels are fixed by the time limit. Physics superscripts attached to stationary radiation conditions can use the opposite sign convention; the relevant pages state the translation.

Prove existence under checkable hypotheses

Cook’s Method gives a complete integrable-interaction argument with a tail error estimate and a three-dimensional dispersive example.

Short-Range Potentials uses momentum localization to reach a broader decay class. It states the completeness theorem and identifies the additional propagation inputs required beyond Cook’s estimate.

Long-Range Potentials proves failure and recovery of asymptotic comparison in an exact logarithmic-phase transport model. It then explains why Coulomb scattering requires a momentum-dependent modifier.

These are sufficient constructions. Failure of one estimate is not a general nonexistence result, and changing the asymptotic reference changes the scattering problem being posed.

Pass to stationary energy data

Stationary Scattering Theory states the limiting absorption principle between weighted spaces and explains the Fourier trace on an energy sphere.

Rigorous Lippmann–Schwinger Equation constructs the outgoing solution for a real bounded compactly supported potential through a local Fredholm equation. The proof shows where radiation uniqueness enters and why invertibility is more general than convergence of a Born series.

Resolvent Methods derives the spectral jump formula, solves a rank-one perturbation, and distinguishes compactness from trace-class scattering conclusions. These calculations also prepare the analytic continuation problem.

Separate resonances from threshold states

Resonances constructs a meromorphic continuation for a three-dimensional compact-support model. Its poles live in a specified momentum or energy sheet, and their interpretation does not override self-adjointness of the physical Hamiltonian.

Complex Scaling gives an exterior-contour construction with explicit transmission conditions. It explains why outgoing modes become square-integrable and which analytic assumptions differ between exterior and global scaling.

Threshold Phenomena distinguishes a regular zero threshold, a resonance, and an eigenstate. A tuned square well and an exactly integrable spectral density show why zero-energy structure affects both stationary limits and time decay.

The articles include solved exercises. Advanced completeness, limiting absorption, unique-continuation, and analytic Fredholm inputs are cited where used; a proof outline is identified as such rather than presented as a complete proof.

References

• G. Teschl, Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators, 2nd ed., Graduate Studies in Mathematics 157, American Mathematical Society, 2014, Chapter 12, author’s text.
• D. Yafaev, “Lectures on scattering theory,” 2004, arXiv:math/0403213.
• M. Zworski, “Mathematical study of scattering resonances,” Bulletin of Mathematical Sciences 7, 1–85, 2017, doi:10.1007/s13373-017-0099-4, author text.