A Critical Quantitative Landis Estimate for the One-Dimensional Quarter-Laplacian

Abstract

We establish a quantitative Landis estimate for the one-dimensional fractional Schrödinger equation $(-\hat{I}”{)}^{1/4}u+V(x)u=0$ in $\mathbb{R}$ with a real-valued bounded potential. If $\Vert V{\Vert }_{L^{\infty }}\le 1$, $\Vert u{\Vert }_{L^{\infty }}\le C_0$, and $\Vert u{\Vert }_{L^2(-1,1)}\ge 1$, then \[ \inf_{|x_0|=R}\|u\|_{L^\infty(x_0-1,x_0+1)} \ge \exp(-CR\log R) \] for all sufficiently large $R$. After the Caffarelli--Silvestre extension and the substitution $y=z^2/2$, the equation becomes a Grushin equation with a weak Robin condition on the degeneracy line. The corresponding angular operator has the arithmetic spectrum $\hat{I}{º}_n=2n+\frac{1}{2}$ after half-density conjugation. The central spectral estimate is \[ \sup_{ξ\in\mathbb R} \bigl\|C\bigl((τ+iξ)^2-L_0\bigr)^{-1}C^*\bigr\| \le Cτ^{-1/2} \] for parameters separated from the angular lattice. It yields a linear-weight Carleman estimate for measurable Robin feedback with absorption threshold $\ddot{I}„\ge C(1+\Vert V{\Vert }_{\infty }^2)$. 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 $C\Vert q{\Vert }_{\infty }^2$ into the global rate $CR\log R$.

38 pages

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