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arXiv · 2503.14888

Perturbed Fourier Transform Associated with Schr\"odinger Operators

Shijun Zheng
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Abstract

We give an exposition on the $L^2$ theory of the perturbed Fourier transform associated with a Schr\"odinger operator $H=-d^2/dx^2 +V$ 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 $\cF$ that covers the class of short range potentials in $(1+|x|)^{-\frac12-\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.

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Shijun Zheng. 2025-03-19. Perturbed Fourier Transform Associated with Schr\"odinger Operators. https://arxiv.org/abs/2503.14888

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