Perturbed Fourier Transform Associated with Schrödinger Operators

We give an exposition on the \(L^2\) theory of the perturbed Fourier transform associated with a Schrödinger operator \(H=-d^2/dx^2 +V\) on the real line, where V  is a real-valued finite measure. In the case \(V\in L^1\cap L^2\) , we explicitly define the perturbed Fourier transform \(\mathcal {F}\) for H and obtain an eigenfunction expansion theorem for square integrable functions. This provides a complete proof of the inversion formula for \({\mathcal F}\) that covers the class of short range potentials in \((1+|x|)^{-\frac 12-\epsilon } L^2 \) . Such paradigm has applications in the study of scattering problems in connection with the spectral properties and asymptotic completeness of the wave operators.