The authors consider the semiclassical Schrödinger operator \[-\hbar\Delta +V(x) - E\] in dimension \(n\neq 2\), where \(\hbar, E > 0\), \(V\) is a real valued potential and satisfies suitable decaying estimates. They are interested in giving resolvent estimates and the main novelty is in case when \(V\) is compactly supported. In particular the resolvent estimates imply logarithmic local energy decay for the wave equation in presence of a compact obstacle (possibly empty) with smooth boundary if the initial data are compactly supported. To prove the theorem the authors adapt a Carleman estimate used in a previous paper of them and suitable weighted functions. The authors also answer a question of K. Datchev and L. Jin [J. Spectr. Theory 10, No. 2, 617–649 (2020; Zbl 1446.35149)].