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 sufficiently fast polynomially decaying small potentials. These are the first known examples, apart from the free