  We prove an explicit weighted estimate for the semiclassical Schr\"odinger
operator $P = - h^2 \partial^2_x + V(x;h)$ on $L^2(\mathbb{R})$, with $V(x;h)$
a finite signed measure, and where $h >0$ is 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 for $P$, and a
limiting absorption resolvent estimate with sharp $h$-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.
