Hölder continuity and Fourier asymptotics of spectral measures for 1D Schrödinger operators under exponentially decaying perturbations

We establish $\frac{1}{2}$-Hölder continuity, or even the Lipschitz property, for the spectral measures of half-line discrete Schrödinger operators under suitable boundary conditions and exponentially decaying small potentials. These are the first known examples, apart from the free case, of Schrödinger operators with Lipschitz continuous spectral measures up to the spectral edge, and it was obtained as a consequence of the Dirichlet boundary condition. Notably, we show that the asymptotic behavior of the time-averaged quantum return probability, either $\log (t)\mathrm{/}t$ or $1\mathrm{/}t$, as in the case of the free Laplacian, remains unchanged in this setting. Furthermore, we prove the persistence of the purely absolutely continuous spectrum and the $\frac{1}{2}$-Hölder continuity of the spectral measures for (Diophantine) quasi-periodic operators under exponentially decaying small perturbations. These results are optimal and hold for all energies, up to the border of the absolutely continuous spectrum.