This chapter is devoted to the peripheral spectral analysis of absorption semigroups generated by formal operators of the form \(M-V\) , where M is the generator of a convolution \(C_{0}\) -semigroup \(\left( \mathcal {M}(t)\right) _{t\ge 0}\) on \(L^{p}(\mathbb {R} ^{d})\) and V is a singular (i.e., unbounded) indefinite potential. We develop a general spectral gap theory and present various applications, including weighted Laplacians on \(\mathbb {R} ^{d}\) , Poincaré inequalities for probability measures, and Witten Laplacians on 1-forms on \(\mathbb {R} ^{d}\) , with further applications to the Helffer–Sjöstrand covariance formula. Notably, the analysis does not rely on any convexity assumptions for the weight functions.