An introduction to the distorted Fourier transform

Abstract This article is intended as an introduction to the distorted Fourier transform associated with a Schrödinger operator on the line or the half-line. This versatile tool has seen numerous applications in nonlinear PDE in recent years. It typically arises in the asymptotic stability analysis of topological solitons in classical field theory, such as kinks in the sine-Gordon or ϕ 4 models. The distorted Fourier transform is also a natural technique in the analysis of dispersive equations on manifolds with symmetries. Such models appear in general relativity, for example in the study of waves on a black-hole background. While microlocal methods have proven to be powerful in such applications, the more classical Weyl–Titchmarsh spectral theory and with it, the distorted Fourier transform, continue to play an essential role in the analysis of evolution PDEs. This article explain how it can be derived from Stone’s formula, which also establishes the Plancherel and inversion theorems.