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

We establish a quantitative Landis estimate for the one-dimensional fractional Schr\"odinger equation $(-\Delta)^{1/4}u+V(x)u=0$ in $\mathbb R$ with a real-valued bounded potential. If $\|V\|_{L^\infty}\le 1$, $\|u\|_{L^\infty}\le C_0$, and $\|u\|_{L^2(-1,1)}\ge 1$, 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 $\kappa_n=2n+\tfrac12$ after half-density conjugation. The central spectral estimate is $$\sup_{\xi\in\mathbb R} \bigl\|C\bigl((\tau+i\xi)^2-L_0\bigr)^{-1}C^*\bigr\| \le C\tau^{-1/2}$$ for parameters separated from the angular lattice. It yields a linear-weight Carleman estimate for measurable Robin feedback with absorption threshold $\tau\ge C(1+\|V\|_\infty^2)$. 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 $C\|q\|_\infty^2$ into the global rate $CR\log R$.