In 1973, Combes and Thomas discovered a general technique for showing exponential decay of eigenfunctions. The technique involved proving the exponential decay of the resolvent of the Schrödinger operator localized between two distant regions. Since then, the technique has been been applied to several types of Schrödinger operators. This dissertation will show that the Combes--Thomas method works well with trace, Hilbert--Schmidt and other trace-type norms. The first result we prove shows exponential decay on trace-type norms of a resolvent of a Schrödinger operator localized between two distant regions. We build on this result by applying the Combes--Thomas method again to prove polynomial and sub-exponential decay estimates on functions of Schrödinger operators localized between two distant regions.