2026/07/30 by Adham Gudaimat
Mathematics · #math.AP #msc:35R11 #msc:35J10 #msc:35J70 #msc:35B60
38 pages
arxiv created 2026/07/30 · arxiv updated 2026/07/31
We establish a quantitative Landis estimate for the one-dimensional fractional Schrödinger equation (-Δ)1/4u+V(x)u=0 in \mathbb R with a real-valued bounded potential. If ‖V‖L^∞≤ 1, ‖u‖L^∞≤ C0, and ‖u‖L2(-1,1)≥ 1, then inf|x0|=R‖u‖L^∞(x0-1,x0+1) ≥ exp(-CRlog R) for all sufficiently large R. After the Caffarelli--Silvestre extension and the substitution y=z2/2, the equation becomes a Grushin equation with a weak Robin condition on the degeneracy line. The corresponding angular operator has the arithmetic spectrum κn=2n+\tfrac12 after half-density conjugation. The central spectral estimate is supξ∈\mathbb R ‖C((τ+iξ)2-L0)-1C^*‖ ≤ Cτ-1/2 for parameters separated from the angular lattice. It yields a linear-weight Carleman estimate for measurable Robin feedback with absorption threshold τ≥ C(1+‖V‖_∞2). Quantitative inward propagation, fixed-scale Grushin-ball propagation, and an interior-cylinder interpolation estimate then transfer bulk non-vanishing to the boundary. The Landis rescaling converts the local potential dependence C‖q‖_∞2 into the global rate CRlog R.