Uniform resolvent estimates for magnetic operators
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917485347864576 |
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| author | D'Ancona, Piero Yin, Zhiqing |
| author_facet | D'Ancona, Piero Yin, Zhiqing |
| contents | We prove Kenig--Ruiz--Sogge type uniform resolvent estimates for selfadjoint magnetic Schrödinger operators $H=(i\partial+A(x))^2+V(x)$ on $\mathbb{R}^{n}$, $n\ge3$. Under suitable decay assumptions on the electric and magnetic potentials, and excluding a threshold resonance at zero, we show that for all $z \in \mathbb{C}\setminus[0,+\infty)$, \begin{equation*}
\|(H-z)^{-1}ϕ\|_{L^{q}}\lesssim|z|^{θ(p,q)}
(1+|z|^γ)
\|ϕ\|_{L^{p}} \end{equation*} throughout the full free resolvent range $(\frac1p,\frac1q)\inΔ(n)$, where $θ(p,q)=\frac n2(\frac1p-\frac1q)-1$. Here $γ=\frac 12\frac{n-1}{n+1}$ under the basic magnetic decay hypothesis, or $γ=\frac{n-1}{4n}$ under a different decay assumption on $A(x)$; for the second case we use a weak endpoint estimate of Frank--Simon type \begin{equation*}
\|R_{0}(z)ϕ\|
_{L^{\frac{2n}{n-1},\infty}_{r}L^{2}_ω}
\lesssim
|z|^{-\frac12}
\|ϕ\|_{L^{\frac{2n}{n+1},1}_{r}L^{2}_ω}. \end{equation*} The result extends the known electromagnetic estimates from fixed frequency and a smaller exponent region to all frequencies and the full Kenig--Ruiz--Sogge range. We also prove a variant with weaker local assumptions in a smaller range $Δ_1(n)$. As applications, we obtain $L^p-L^{p'}$ restriction type estimates for the density of the spectral measure of magnetic Schrödinger operators, and an eigenvalue enclosure result for complex scalar perturbations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_11151 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Uniform resolvent estimates for magnetic operators D'Ancona, Piero Yin, Zhiqing Analysis of PDEs Mathematical Physics 35J10, 35P25, 42B37 We prove Kenig--Ruiz--Sogge type uniform resolvent estimates for selfadjoint magnetic Schrödinger operators $H=(i\partial+A(x))^2+V(x)$ on $\mathbb{R}^{n}$, $n\ge3$. Under suitable decay assumptions on the electric and magnetic potentials, and excluding a threshold resonance at zero, we show that for all $z \in \mathbb{C}\setminus[0,+\infty)$, \begin{equation*} \|(H-z)^{-1}ϕ\|_{L^{q}}\lesssim|z|^{θ(p,q)} (1+|z|^γ) \|ϕ\|_{L^{p}} \end{equation*} throughout the full free resolvent range $(\frac1p,\frac1q)\inΔ(n)$, where $θ(p,q)=\frac n2(\frac1p-\frac1q)-1$. Here $γ=\frac 12\frac{n-1}{n+1}$ under the basic magnetic decay hypothesis, or $γ=\frac{n-1}{4n}$ under a different decay assumption on $A(x)$; for the second case we use a weak endpoint estimate of Frank--Simon type \begin{equation*} \|R_{0}(z)ϕ\| _{L^{\frac{2n}{n-1},\infty}_{r}L^{2}_ω} \lesssim |z|^{-\frac12} \|ϕ\|_{L^{\frac{2n}{n+1},1}_{r}L^{2}_ω}. \end{equation*} The result extends the known electromagnetic estimates from fixed frequency and a smaller exponent region to all frequencies and the full Kenig--Ruiz--Sogge range. We also prove a variant with weaker local assumptions in a smaller range $Δ_1(n)$. As applications, we obtain $L^p-L^{p'}$ restriction type estimates for the density of the spectral measure of magnetic Schrödinger operators, and an eigenvalue enclosure result for complex scalar perturbations. |
| title | Uniform resolvent estimates for magnetic operators |
| topic | Analysis of PDEs Mathematical Physics 35J10, 35P25, 42B37 |
| url | https://arxiv.org/abs/2504.11151 |