Christian Remling Some Schr\"odinger operators with power-decaying potentials and pure point spectrum (38K, LaTeX) ABSTRACT. We construct deterministic (= non-random) potentials $V(x)=O(x^{-c})$ such that the one-dimensional Schr\"odinger equation $-y''+Vy=Ey$ on the half-axis $x\in [0,\infty)$ has dense pure point spectrum in $(0,\infty)$ for almost all boundary conditions at $x=0$. A modification of this construction yields power-decaying potentials for which the spectrum is purely singular continuous in $(0,\infty)$ for all boundary conditions.