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 $H=-d^2\mathrm{/}d{x}^2+V$ on the real line, where $V$ is a real-valued \mbox{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 $\text{cF}$ that covers the class of short range potentials in $(1+\mathrm{\mid }x\mathrm{\mid }{)}^{-\frac{1}{2}-\text{eps}}{L}^2$. Such paradigm has applications in the study of scattering problems in connection with the spectral properties and asymptotic completeness of the wave operators.