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We prove an explicit weighted estimate for the semiclassical Schrödinger operatorP = - h^{2} \partial^{2}_{x} + V(x;h)onL^{2}(\R), withV(x;h)a finite signed measure, and whereh >0is the semiclassical parameter. The proof is a one-dimensional instance of the spherical energy method, which has been used to prove Carleman estimates in higher dimensions and in more complicated geometries. The novelty of our result is that the potential need not be absolutely continuous with respect to Lebesgue measure. Two consequences of the weighted estimate are the absence of positive eigenvalues forP, and a limiting absorption resolvent estimate with sharph-dependence. The resolvent estimate implies exponential time-decay of the local energy for solutions to the corresponding wave equation with a compactly supported measure potential, provided there are no negative eigenvalues and no zero resonance, and provided the initial data have compact support.