In this work, we investigate the compactness and the long time behavior of killed Feynman-Kac semigroups of various processes arising from statistical physics with very general singular Schrödinger potentials. The processes we consider cover a large class of processes used in statistical physics, with strong links with quantum mechanics and (local or not) Schrödinger operators (including e.g. fractional Laplacians). For instance we consider solutions to elliptic differential equations, Lévy processes, the kinetic Langevin process with locally Lipschitz gradient fields, and systems of interacting Lévy particles. Our analysis relies on a Perron-Frobenius type theorem derived in a previous work [A. Guillin, B. Nectoux, L. Wu, 2020 J. Eur. Math. Soc.] for Feller kernels and on the tools introduced in [L. Wu, 2004, Probab. Theory Relat. Fields] to compute bounds on the essential spectral radius of a bounded nonnegative kernel.