Perturbed Fourier Transform Associated with Schrödinger Operators
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912282530807808 |
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| author | Zheng, Shijun |
| author_facet | Zheng, Shijun |
| contents | 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 \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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_14888 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Perturbed Fourier Transform Associated with Schrödinger Operators Zheng, Shijun Analysis of PDEs Mathematical Physics 35P25, 35J10 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 \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. |
| title | Perturbed Fourier Transform Associated with Schrödinger Operators |
| topic | Analysis of PDEs Mathematical Physics 35P25, 35J10 |
| url | https://arxiv.org/abs/2503.14888 |