2025/04/15 by Piero DʼAncona, D'Ancona, Piero, Zhiqing Yin +1
Mathematics · #35J10 #35P25 #42B37 #Advanced Harmonic Analysis Research #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2504.11151
openalex publication_date 2025/04/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove Kenig--Ruiz--Sogge type uniform resolvent estimates for selfadjoint magnetic Schrödinger operators H=(i∂+A(x))2+V(x) on ℝn, n≥3. Under suitable decay assumptions on the electric and magnetic potentials, and excluding a threshold resonance at zero, we show that for all z ∈ ℂ∖[0,+∞), ‖(H-z)-1ϕ‖Lq\lesssim|z|θ(p,q) (1+|z|γ) ‖ϕ‖Lp throughout the full free resolvent range (\frac1p,\frac1q)∈Δ(n), where θ(p,q)=\frac n2(\frac1p-\frac1q)-1. Here γ=\frac 12(n-1)/(n+1) under the basic magnetic decay hypothesis, or γ=(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 ‖R0(z)ϕ‖ _L(2n)/(n-1),∞rL2ω \lesssim |z|-\frac12 ‖ϕ‖_L(2n)/(n+1),1rL2ω. 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 Δ1(n). As applications, we obtain Lp-Lp' restriction type estimates for the density of the spectral measure of magnetic Schrödinger operators, and an eigenvalue enclosure result for complex scalar perturbations.