Perturbed Fourier Transform Associated with Schr\"odinger Operators

We give an exposition on the L² theory of the perturbed Fourier transform associated with a Schr\"odinger operator H=-d²/dx² +V on the real line, where V is a real-valued finite measure. In the case V∈ L¹∩ L², we explicitly define the perturbed Fourier transform F for H and obtain an eigenfunction expansion theorem for square integrable functions. This provides a complete proof of the inversion formula for that covers the class of short range potentials in (1+|x|)⁻¹²⁻ L². Such paradigm has applications in the study of scattering problems in connection with the spectral properties and asymptotic completeness of the wave operators.